Swiss Ephemeris is a function
package of astronomical calculations that serves the needs of astrologers,
archaeoastronomers, and, depending on purpose, also the needs of astronomers.
It includes long-term ephemerides for the Sun, the Moon, the planets, more than
300’000 asteroids, historically relevant fixed stars and several ”hypothetical”
objects.

The precision of the Swiss Ephemeris is at least as good as that
of the Astromical Almanac, which follows current standards of ephemeris
calculation. Swiss Ephemeris will, as we hope, be able to
keep abreast to the scientific advances in ephemeris computation for the coming
decades.

The Swiss Ephemeris package consists of source
code in C, a DLL, a collection of ephemeris files and a few sample programs
which demonstrate the use of the DLL and the Swiss Ephemeris graphical label.
The ephemeris files contain compressed astronomical ephemerides

Full C source code is included with the Swiss Ephemeris, so that
non-Windows programmers can create a linkable or shared library in their
environment and use it with their applications.

The Swiss Ephemeris is not a product for end users. It is a toolset for
programmers to build into their astrological software.

Swiss Ephemeris is made available by its authors under a dual licensing system. The software developer, who uses any
part of Swiss Ephemerisin his or her
software, must choose between one of the two license models, which are

a) GNU public license version 2
or later

b) Swiss Ephemeris Professional
License

The choice must be made before the software developer distributes
software containing parts of Swiss Ephemeris to others, and before any public
service using the developed software is activated.

If the developer choses the GNU GPL software license, he or she must
fulfill the conditions of that license, which includes the obligation to place
hisor her whole software project under
the GNU GPL or a compatible license.See
http://www.gnu.org/licenses/old-licenses/gpl-2.0.html

If the developer choses the Swiss Ephemeris Professional license,he must follow the instructions as found in
http://www.astro.com/swisseph/and
purchase the Swiss Ephemeris Professional Edition from Astrodienst and sign the
corresponding license contract.

The Swiss Ephemeris Professional Edition can be purchased from
Astrodienst for a one-time fixed fee for each commercial programming
project. The license is just a legal document. All actual software and data are
found in the public download area and are to be downloaded from there.

Professional license: The license fee for the first
license is Swiss Francs (CHF) 750.- , and CHF 400.-for each additional license by the same
licensee. An unlimited license is available for CHF 1550.-.

The core part of Swiss Ephemeris is a compression of the JPL-Ephemeris DE431,
which covers roughly the time range 13’000 BCE to 17’000 CE.Using a sophisticated mechanism, we succeeded
in reducing JPL's 2.8 GB storage to only 99 MB. The compressed version agrees
with the JPL Ephemeris to 1 milli-arcsecond (0.001”).Since the inherent uncertainty of the JPL
ephemeris for most of its time range is a lot greater, the Swiss Ephemeris
should be completely satisfying even for computations demanding very high
accuracy.

(Before 2014, the Swiss Ephemeris was based on JPL Ephemeris DE406. Its
200 MB were compressed to 18 MB. The time range of the DE406 was 3000 BC to
3000 AD or 6000 years. We had extended this time range to 10'800 years,
from 2 Jan 5401 BC to 31 Dec 5399. The details of this extension are described
below in section 2.1.5. To make sure that you work with current data, please
check the date of the ephemeris files. They must be 2014 or later.)

Each Swiss Ephemeris file covers a period of 600 years; there are 50
planetary files, 50 Moon files for the whole time range of almost 30’000 years
and 18 main-asteroid files for the time range of 10'800 years.

The file names are as follows:

Planetary file

Moon file

Main asteroid file

Time range

Seplm132.se1

Semom132.se1

11 Aug 13000 BC – 12602 BC

Seplm126.se1

Semom126.se1

12601 BC – 12002 BC

Seplm120.se1

Semom120.se1

12001 BC – 11402 BC

Seplm114.se1

Semom114.se1

11401 BC – 10802 BC

Seplm108.se1

Semom108.se1

10801 BC – 10202 BC

Seplm102.se1

Semom102.se1

10201 BC – 9602 BC

Seplm96.se1

Semom96.se1

9601 BC – 9002 BC

Seplm90.se1

Semom90.se1

9001 BC – 8402 BC

Seplm84.se1

Semom84.se1

8401 BC – 7802 BC

Seplm78.se1

Semom78.se1

7801 BC – 7202 BC

Seplm72.se1

Semom72.se1

7201 BC – 6602 BC

Seplm66.se1

Semom66.se1

6601 BC – 6002 BC

Seplm60.se1

Semom60.se1

6001 BC – 5402 BC

seplm54.se1

semom54.se1

seasm54.se1

5401 BC – 4802 BC

seplm48.se1

semom48.se1

seasm48.se1

4801 BC – 4202 BC

seplm42.se1

semom42.se1

seasm42.se1

4201 BC – 3602 BC

seplm36.se1

semom36.se1

seasm36.se1

3601 BC – 3002 BC

seplm30.se1

semom30.se1

seasm30.se1

3001 BC – 2402 BC

seplm24.se1

semom24.se1

seasm24.se1

2401 BC – 1802 BC

seplm18.se1

semom18.se1

seasm18.se1

1801 BC – 1202 BC

seplm12.se1

semom12.se1

seasm12.se1

1201 BC – 602 BC

seplm06.se1

semom06.se1

seasm06.se1

601 BC – 2 BC

sepl_00.se1

semo_00.se1

seas_00.se1

1 BC – 599 AD

sepl_06.se1

semo_06.se1

seas_06.se1

600 AD – 1199 AD

sepl_12.se1

semo_12.se1

seas_12.se1

1200 AD – 1799 AD

sepl_18.se1

semo_18.se1

seas_18.se1

1800 AD – 2399 AD

sepl_24.se1

semo_24.se1

seas_24.se1

2400 AD – 2999 AD

sepl_30.se1

semo_30.se1

seas_30.se1

3000 AD – 3599 AD

sepl_36.se1

semo_36.se1

seas_36.se1

3600 AD – 4199 AD

sepl_42.se1

semo_42.se1

seas_42.se1

4200 AD – 4799 AD

sepl_48.se1

semo_48.se1

seas_48.se1

4800 AD – 5399 AD

sepl_54.se1

semo_54.se1

5400 AD – 5999 AD

sepl_60.se1

semo_60.se1

6000 AD – 6599 AD

sepl_66.se1

semo_66.se1

6600 AD – 7199 AD

sepl_72.se1

semo_72.se1

7200 AD – 7799 AD

sepl_78.se1

semo_78.se1

7800 AD – 8399 AD

sepl_84.se1

semo_84.se1

8400 AD – 8999 AD

sepl_90.se1

semo_90.se1

9000 AD – 9599 AD

sepl_96.se1

semo_96.se1

9600 AD – 10199 AD

sepl_102.se1

semo_102.se1

10200 AD – 10799 AD

sepl_108.se1

semo_108.se1

10800 AD – 11399 AD

sepl_114.se1

semo_114.se1

11400 AD – 11999 AD

sepl_120.se1

semo_120.se1

12000 AD – 12599 AD

sepl_126.se1

semo_126.se1

12600 AD – 13199 AD

sepl_132.se1

semo_132.se1

13200 AD – 13799 AD

sepl_138.se1

semo_138.se1

13800 AD – 14399 AD

sepl_144.se1

semo_144.se1

14400 AD – 14999 AD

sepl_150.se1

semo_150.se1

15000 AD – 15599 AD

sepl_156.se1

semo_156.se1

15600 AD – 16199 AD

sepl_162.se1

semo_162.se1

16200 AD – 7 Jan 16800 AD

All Swiss Ephemeris files have the file suffix .se1.

A planetary file is about500 kb,
a lunar file 1300 kb.

Swiss Ephemeris files are available for download from Astrodienst's web
server.

The time range of the Swiss Ephemeris

Versions until 1.80, which
were based on JPL Ephemeris DE406 and some extension created by Astrodienst,
work for the following time range:

Start date2 Jan 5401 BC(-5400) jul.=
JD-251291.5

End date31
Dec 5399 AD (greg. Cal.) = JD
3693368.5

Versions since 2.00, which
are based on JPL Ephemeris DE431, work for the following time range:

Start date 11 Aug 13000 BCE (-12999) jul.= JD -3026604.5

End date7 Jan 16800 CE greg.= JD 7857139.5

Please note that versions
prior to 2.00 are not able to correctly handle the JPL ephemeris
DE431.

A note
on year numbering:

There are
two numbering systems for years before the year 1 AD. The historical numbering
system (indicated with BC) has no year zero. Year 1 BC is followed directly by
year 1 AD.

The
astronomical year numbering system does have a year zero; years before the
common era are indicated by negative year numbers. The sequence is year -1,
year 0, year 1 AD.

The
historical year 1 BC corresponds to astronomical year 0,

the
historical your 2 BC corresponds to astronomical year -1, etc.

In this
document and other documents related to the Swiss Ephemeris we use both systems
of year numbering. When we write a negative year number, it is astronomical
style; when we write BC, it is historical style.

This is a semi-analytical approximation of the JPL planetary and lunar
ephemerides DE404, developed by Steve Moshier. Its deviation from JPL is below
1 arc second with the planets and a few arc seconds with the moon. No data
files are required for this ephemeris, as all data are linked into the
program code already.

This may be sufficient accuracy for most purposes, since the moon moves
1 arc second in 2 time seconds and the sun 2.5 arc seconds in one minute.

The advantage of the Moshier mode of the Swiss Ephemeris is that it
needs no disk storage. Its disadvantage besides the limited precision is
reduced speed: it is about 10 times slower than JPL mode and the compressed JPL
mode (described above).

The Moshier Ephemeris covers the interval from 3000 BC to 3000 AD.
However, Moshier notes that “the adjustment for the inner planets is strictly
valid only from 1350 B.C. to 3000 A.D., but may be used to 3000 B.C. with some
loss of precision”. And:“The Moon's
position is calculated by a modified version of the lunar theory of
Chapront-Touze' and Chapront. This has a precision of 0.5 arc second relative
to DE404 for all dates between 1369 B.C. and 3000 A.D. “ (Moshier,
http://www.moshier.net/aadoc.html).

This is the full precision state-of-the-art ephemeris. It provides the
highest precision and is the basis of the Astronomical Almanac. Time range:

Start date 9 Dec 13002 BCE (-13001) jul.= JD -3027215.5

End date11 Jan 17000 CE greg.= JD 7930192.5

JPL is the Jet Propulsion Laboratory of NASA in Pasadena, CA, USA (see http://www.jpl.nasa.gov ). Since many years this
institute which is in charge of the planetary missions of NASA has been the source
of the highest precision planetary ephemerides. The currently newest version of
JPL ephemeris is the DE430/DE431.

There are several versions of the JPL Ephemeris. The version is
indicated by the DE-number. A higher number indicates a more recent version.
SWISSEPH should be able to read any JPL file from DE200 upwards.

Accuracy of JPL
ephemerides DE403/404 (1996) and DE405/406 (1998)

According to a paper (see below) by Standish and others on DE403 (of
which DE406 is only a slight refinement), the accuracy of this ephemeris can be
partly estimated from its difference from DE200:

With the inner planets, Standish shows that within the period
1600 – 2160 there is a maximum difference of 0.1 – 0.2” which is mainly due to
a mean motion error of DE200. This means that the absolute precision of DE406
is estimated significantly better than 0.1” over that period. However, for the
period 1980 – 2000 the deviations between DE200 and DE406 are below 0.01” for all
planets, and for this period the JPL integration has been fit to measurements
by radar and laser interferometry, which are extremely precise.

With the outer planets, Standish's diagrams show that there are
large differences of several ” around 1600, and he says that these deviations
are due to the inherent uncertainty of extrapolating the orbits beyond the
period of accurate observational data.The uncertainty of Pluto exceeds
1” before 1910 and after 2010, and increases rapidly in more remote past or
future.

With the moon, there is an increasing difference of 0.9”/cty2between 1750
and 2169. It is mainly causedby errors
in LE200 (Lunar Ephemeris).

The differences between DE200 and DE403 (DE406) can be summarized as
follows:

The Swiss Ephemeris is based on the latest JPL
file, and reproduces the full JPL precision with better than 1/1000 of an arc
second, while requiring only a tenth storage. Therefore for most applications
it makes little sense to get the full JPL file. Precision comparison can be
done at the Astrodienst web server. The Swiss Ephemeris test page http://www.astro.com/swisseph/swetest.htm
allows to compute planetary positions for any date using the full JPL
ephemerides DE200, DE406, DE421, DE431, or the compressed Swiss Ephemeris or
the Moshier ephemeris.

The original JPL ephemeris gives barycentric equatorial Cartesian
positions relative to the equinox 2000. Moshier provides heliocentric
positions.The conversions to apparent
geocentric ecliptical positions were done using the algorithms and constants of
the Astronomical Almanac as described in the ”Explanatory Supplement to the
Astronomical Almanac”. Using the DE200 data file, it is possible to reproduce
the positions given by the Astronomical Almanac 1995, 1996, and 1997 to the
last digit. Editions of other years have not been checked.

Since 2003, the Astronomical Almanac has been using JPL ephemeris DE405,
and since Astronomical Almanac 2006 all relevant resolutions of the
International Astronomical Union (IAU) have been implemented. Versions 1.70 and
higher of the Swiss Ephemeris also follow these resolutions and reproduce the
sample calculation given by AA2006 (p. B61-B63), AA2011 and AA2013 (both p.
B68-B70) to the last digit, i.e. to better than 0.001 arc second. (To avoid
confusion when checking AA2006, it may be useful to know that the JD given on
page B62 does not have enough digits in order to produce the correct final
result. With later AA2011 and AA2013, there is no such problem.)

The Swiss Ephemeris, from version 1.70 on, reproduces astrometric planetary positions of the
JPL Horizons System precisely. However, there are small differences of about 52
mas (milli-arcseconds) with apparent
positions. The same deviations also occur if Horizons is compared with the
example calculations given in the Astronomical Almanac.

Horizons uses an entirely different approach and a different reference
system. It follows IERS Conventions 1996 (p. 22), i. e. it uses the old
precession models IAU 1976 (Lieske) and nutation IAU 1980 (Wahr) and corrects
the resulting positions by adding daily-measured celestial pole offsets
(delta_psi and delta_epsilon) to nutation.

On the other hand, the Astronomical Almanac and the Swiss Ephemeris
follow IERS Conventions 2003 and 2010, but do not take into account daily
celestial pole offsets.

While Horizons’ approach is more accurate in that it takes into account
very small and unpredictable motions of the celestial pole (free core
nutation), the resulting positions are not relative to the same reference frame
as Astronomical Almanac and the Swiss Ephemeris, and they are not in agreement
with the recent IERS Conventions 2003 and 2010. Some component of so-called
frame bias is lost in Horizons positions. This causes a more or less constant
offset of 52 mas in right ascension or 42 mas in ecliptic longitude.

Swiss Ephemeris versions 2.00 and higher contain code to reproduce
positions of Horizons with a precision of about 1 mas for 1799 AD – today.
Before 1799, the deviations in apparent positions between the Swiss Ephemeris
and Horizons slowly increase. This is explained by the fact that Horizons uses
the long-term precession model Owen 1990 for the remote past and future,
whereas the Swiss Ephemeris uses the long-term precession model Vondrák 2011.

For best agreement with Horizons, current data files with earth
orientation parameters (EOP) must be downloaded from the IERS website and put
into the ephemeris path. If they are not available, the Swiss Ephemeris uses an
approximation which reproduces Horizons still with an accuracy of about 2 mas
between 1962 and present.

It must be noted that correct values for delta_psi and delta_epsilon are
only available between 1962 and present. For all calculations before that,
Horizons uses the first values of the EOP data, and for all calculations in the
future, it uses the last values of the existing data are used. The resulting
positions are not really correct, but the ephemeris is at least continuous.

More information on this and technical details are found in the
programmer’s documentation and in the source code, file swephlib.h.

IERS Conventions 1996, 2003, and 2010 can be read or downloaded from
here:

http://www.iers.org/IERS/EN/DataProducts/Conventions/conventions.html

Many thanks to Jon Giorgini, developer of the Horizons System, for
explaining us the methods used at JPL.

With version 1.70, the standard algorithms recommended by the IAU
resolutions up to 2005 were implemented. The following calculations have been
added or changed with Swiss Ephemeris version 1.70:

- "Frame Bias" transformation from ICRS to J2000.

- Nutation IAU 2000B (could be switched to 2000A by the user)

- Precession model P03 (Capitaine/Wallace/Chapront 2003), including
improvements in ecliptic obliquity and sidereal time that were achieved by this
model

The differences between the old and new planetary positions in ecliptic longitude (arc seconds) are:

yearnew - old

2000-0.00108

19950.02448

19800.05868

19700.10224

19500.15768

19000.30852

18000.58428

1799-0.04644

1700-0.07524

1500-0.12636

1000-0.25344

0-0.53316

-1000-0.85824

-2000-1.40796

-3000-3.33684

-4000-10.64808

-5000-32.68944

-5400-49.15188

The discontinuity of the curve between 1800 and 1799 is explained by the
fact that old versions of the Swiss Ephemeris used different precession models
for different time ranges: the model IAU 1976 by Lieske for 1800 - 2200, and
the precession model by Williams 1994 outside that time range.

Note: Precession model P03 is said to be accurate to 0.00005 arc second
for CE 1000-3000.

The differences between version 1.70 and older versions for the future
are as follows:

2000-0.00108

2010-0.01620

2050-0.14004

2100-0.29448

2200-0.61452

22010.05940

30000.27252

40000.48708

50000.47592

54000.40032

The discontinuity in 2200 has the same explanation as
the one in 1800.

Jyotish / sidereal
ephemerides:

The ephemeris changes by a constant value of about +0.3 arc second. This
is because all our ayanamsas have the start epoch 1900, for which epoch
precession was corrected by the same amount.

Former versions of the Swiss Ephemeris had used the precession model by
Capitaine, Wallace, and Chapront of 2003 for the time range 1800-2200 and the
precession model J. G. Williams in Astron. J. 108, 711-724 (1994) for epochs
outside this time range.

Version 1.78 calculates precession and ecliptic obliquity according to
Vondrák, Capitaine, and Wallace, “New precession expressions, valid for long
time intervals”, A&A 534, A22 (2011), which is good for +- 200 millennia.

This change has almost no ramifications for historical epochs. Planetary
positions and the obliquity of the ecliptic change by less than an arc minute
in 5400 BC. However, for research concerning the prehistoric cave paintings
(Lascaux, Altamira, etc, some of which may represent celestial constellations),
fixed star positions are required for 15’000 BC or even earlier (the Chauvet
cave was painted in 33’000 BC). Such calculations are now possible using the
Swiss Ephemeris version 1.78 or higher. However, the Sun, Moon, and the planets
remain restricted to the time range 5400 BC to 5400 AD.

These differences are explained by the fact that the Swiss Ephemeris is
now based on JPL Ephemeris DE431, whereas before release 2.00 it was based on
DE406. The differences are listed above in ch. 2.1.1.3, see paragraph on “Comparison of JPL ephemerides DE406 (1998) with
DE431 (2013)”.

The following conversions are applied to the coordinates after reading
the raw positions from the ephemeris files:

Correction for light-time. Since the planet's light
needs time to reach the earth, it is never seen where it actually is, but where
it was some time before. Light-time amounts to a few minutes with the inner
planets and a few hours with distant planets like Uranus, Neptune and Pluto.
For the moon, the light-time correction is about one second. With planets,
light-time correction may be of the order of 20” in position, with the moon
0.5”

Conversion from the solar system barycenter to the
geocenter. Original JPL data are referred to the center of the gravity of the
solar system. Apparent planetary positions are referred to an imaginary
observer in the center of the earth.

Light deflection by the gravity of the sun. In the
gravitational fields of the sun and the planets light rays are bent. However,
within the solar system only the sun has enough mass to deflect light
significantly. Gravity deflection is greatest for distant planets and stars,
but never greater than 1.8”. When a planet disappears behind the sun, the Explanatory
Supplement recommends to set the deflection = 0. To avoid discontinuities,
we chose a different procedure. See Appendix A.

”Annual” aberration of light. The velocity of
light is finite, and therefore the apparent direction of a moving body from a
moving observer is never the same as it would be if both the planet and the
observer stood still. For comparison: if you run through the rain, the rain
seems to come from ahead even though it actually comes from above. Aberration
may reach 20”.

Frame Bias (ICRS to J2000). JPL ephemeredes since
DE403/DE404 are referred to the International Celestial Reference System, a
time-independent, non-rotating reference system which was introduced by the IAU
in 1997. The planetary positions and speed vectors are rotated to the J2000
system. This transformation makes a difference of only about 0.0068 arc seconds
in right ascension. (Implemented from Swiss Ephemeris 1.70 on)

Precession. Precession is the motion of
the vernal equinox on the ecliptic. It results from the gravitational pull of
the Sun, the Moon, and the planets on the equatorial bulge of the earth.
Original JPL data are referred to the mean equinox of the year 2000. Apparent
planetary positions are referred to the equinox of date. (From Swiss
Ephemeris 1.78 on, we use the precession model Vondrák/Capitaine/Wallace 2011.)

Nutation (true equinox of date).
A short-period oscillation of the vernal equinox. It results from the moon’s
gravity which acts on the equatorial bulge of the earth. The period of nutation
is identical to the period of a cycle of the lunar node, i.e. 18.6 years. The
difference between the true vernal point and the mean one is always below 17”.
(From Swiss Ephemeris 2.00, we use the nutation model IAU 2006. Since 1.70, we
used nutation model IAU 2000. Older versions used the nutation model IAU 1980
(Wahr).)

Transformation from equatorial to ecliptic coordinates

For precise speed of the planets and the moon, we had to make a
special effort, because the Explanatory Supplement does not give
algorithms that apply the above-mentioned transformations to speed. Since this
is not a trivial job, the easiest way would have been to compute three
positions in a small interval and determine the speed from the derivation of
the parabola going through them. However, double float calculation does not
guarantee a precision better than 0.1”/day. Depending on the time difference
between the positions, speed is either good near station or during fast motion.
Derivation from more positions and higher order polynomials would not help
either.

Therefore we worked out a way to apply directly all the transformations
to the barycentric speeds that can be derived from JPL or Swiss Ephemeris. The
precision of daily motion is now better than 0.002” for all planets, and the
computation is even a lot faster than it would have been from three positions.
A position with speed takes in average only 1.66 times longer than one without
speed, if a JPL or a Swiss Ephemeris position is computed. With Moshier,
however, a computation with speed takes 2.5 times longer.

The idea behind our mechanism of ephemeris compression was developed by
Dr. Peter Kammeyer of the U.S. Naval Observatory in 1987.

This is how it works: The ephemerides of the Moon and the inner planets
require by far the greatest part of the storage. A more sophisticated mechanism
is required for these than for the outer planets.Instead of the positions we store the
differences between JPL and the mean orbits of the analytical theory VSOP87.
These differences are a lot smaller than the position values, wherefore they
require less storage.They are stored in
Chebyshew polynomials covering a period of an anomalistic cycle each. (By the
way, this is the reason, why the Swiss Ephemeris does not cover the time range
of the full JPL ephemeris. The first ephemeris file begins on the date on which
the last of the inner planets (including Mars) passes its first perihelion
after the start date of the JPL ephemeris.)

With the outer planets from Jupiter through Pluto we use a simpler
mechanism. We rotate the positions provided by the JPL ephemeris to the mean
plane of the planet. This has the advantage that only two coordinates have high
values, whereas the third one becomes very small. The data are stored in
Chebyshew polynomials that cover a period of 4000 days each.(This is the reason, why Swiss Ephemeris
stops before the end date of the JPL ephemeris.)

This chapter is only relevant for those who use pre-2014, DE406-based
ephemeris files of the Swiss Ephemeris.

The JPL ephemeris DE406 covers the time range from 3000 BC to 3000 AD.
While this is an excellent range covering all precisely known historical
events, there are some types of ancient astrology and archaeoastronomical
research which would require a longer time range.

In December 1998 we have made an effort to extend the time range using
our own numerical integration. The exact physical model used by Standish et.
al. for the numerical integration of the DE406 ephemeris is not fully
documented (at least we do not understand some details), so that we cannot use
the same integration program as had been used at JPL for the creation of the
original ephemeris.

The previous JPL ephemeris DE200, however, has been reproduced by Steve
Moshier over a very long time range with his numerical integrator, which was
available to us. We used this software with start vectors taken at the end
points of the DE406 time range. To test our numerical integrator, we ran it
upwards from 3000 BC to 600 BC for a period of 2400 years and compared its
results with the DE406 ephemeris itself. The agreement is excellent for all
planets except the Moon (see table below). The lunar orbit creates a problem
because the physical model for the Moon's libration and the effect of the tides
on lunar motion is quite different in the DE406 from the model in the DE200. We
varied the tidal coupling parameter (love number) and the longitudinal
libration phase at the start epoch until we found the best agreement over the
2400 year test range between our integration and the JPL data. We could
reproduce the Moon's motion over a the 2400 time range with a maximum error of
12 arcseconds. For most of this time range the agreement is better than 5
arcsec.

With these modified parameters we ran the integration backward in time
from 3000 BC to 5400 BC. It is reasonable to assume that the integration errors
in the backward integration are not significantly different from the
integration errors in the upward integration.

Planet

max. Error arcsec

avg. error arcec

Mercury

1.67

0.61

Venus

0.14

0.03

Earth

1.00

0.42

Mars

0.21

0.06

Jupiter

0.85

0.38

Saturn

0.59

0.24

Uranus

0.20

0.09

Neptune

0.12

0.06

Pluto

0.12

0.04

Moon

12.2

2.53

Sun bary.

6.3

0.39

The same procedure was applied
at the upper end of the DE406 range, to cover an extension period from 3000 AD
to 5400 AD. The maximum integration errors as determined in the test run 3000
AD down to 600 AD are given in the table below.

JPL ephemerides do not include a mean lunar node or mean lunar apsis
(perigee/apogee). We therefore have to derive them from different sources.

Our mean node and mean apogee are computed from Moshier's lunar routine,
which is an adjustment of the ELP2000-85 lunar theory to the JPL ephemeris on
the interval from 3000 BC to 3000 AD. Its deviation from the mean node of
ELP2000-85 is 0 for J2000 and remains below 20 arc seconds for the whole
period. With the apogee, the deviation reaches 3 arc minutes at 3000 BC.

In order to cover the whole time range of DE431, we had to add some
corrections to Moshier’s mean node and apsis, which we derived from the true
node and apsis that result from the DE431 lunar ephemeris. Estimated precision
is 1 arcsec, relative to DE431.

Notes for
Astrologers:

Astrological Lilith or the Dark
Moon is either the apogee (”aphelion”) of the lunar orbital ellipse or,
according to some, its empty focal point.As seen from the geocenter, this makes no difference. Both of them are
located in exactly the same direction. But the definition makes a difference
for topocentric ephemerides.

The opposite point, the lunar perigee or orbital point closest to the
Earth, is also known as Priapus. However, if Lilith is understood as the
second focal point, an opposite point makes no sense, of course.

Originally, the term ”Dark Moon” stood for a hypothetical second body
that was believed to move around the earth. There are still ephemerides
circulating for such a body, but modern celestial mechanics clearly exclude the
possibility of such an object. Later the term ”Dark Moon” was used for the
lunar apogee.

The Swiss Ephemeris apogee differs from the ephemeris given by Joëlle de
Gravelaine in her book ”Lilith, der schwarze Mond” (Astrodata 1990). The
difference reaches several arc minutes. The mean apogee (or perigee) moves
along the mean lunar orbit which has an inclination of 5 degrees. Therefore it
has to be projected on the ecliptic. With de Gravelaine's ephemeris, this was
not taken into account. As a result of this projection, we also provide an
ecliptic latitude of the apogee, which will be of importance if declinations
are used.

There may be still another problem. The 'first' focal point does not
coincide with the geocenter but with the barycenter of the earth-moon-system.
The difference is about 4700 km. If one took this into account, it would result
in a monthly oscillation of the Black Moon. If one defines the Black Moon as
the apogee, this oscillation would be about +/- 40 arc minutes. If one defines
it as the second focus, the effect is a lot greater: +/- 6 degrees. However, we
have neglected this effect.

[added by Alois 7-feb-2005, arising out of a discussion with Juan
Revilla] The concept of 'mean lunar orbit' means that short term. e.g. monthly,
fluctuations must not be taken into account. In the temporal average, the EMB
coincides with the geocenter. Therefore, when mean elements are computed, it is
correct only to consider the geocenter, not the Earth-Moon Barycenter.

Computing topocentric positions of mean elements is also meaningless and
should not be done.

The 'true' lunar node is usually considered the osculating node element
of the momentary lunar orbit. I.e., the axis of the lunar nodes is the
intersection line of the momentary orbital plane of the moon and the plane of
the ecliptic. Or in other words, the nodes are the intersections of the two
great circles representing the momentary apparent orbit of the moon and the
ecliptic.

The nodes are considered important because they are connected with
eclipses. They are the meeting points of the sun and the moon. From this point
of view, a more correct definition might be: The axis of the lunar nodes is the
intersection line of the momentary orbital plane of the moon and the
momentary orbital plane of the sun.

This makes a difference, although a small one. Because of the monthly
motion of the earth around the earth-moon barycenter, the sun is not exactly on
the ecliptic but has a latitude, which, however, is always below an arc second.
Therefore the momentary plane of the sun's motion is not identical with the
ecliptic. For the true node, this would result in a difference in longitude of
several arc seconds.However, Swiss
Ephemeris computes the traditional version.

The advantage of the 'true' nodes against the mean ones is that when the
moon is in exact conjunction with them, it has indeed a zero latitude. This is
not so with the mean nodes.

In the strict sense of the word, even the ”true” nodes are true only
twice a month, viz. at the times when the moon crosses the ecliptic. Positions
given for the times in between those two points are based on the idea that
celestial orbits can be approximated by elliptical elements or great circles.
The monthly oscillation of the node is explained by the strong perturbation of
the lunar orbit by the sun. A different approach for the “true” node that would
make sense, would be to interpolate between the true node passages. The monthly
oscillation of the node would be suppressed, and the maximum deviation from the
conventional ”true” node would be about 20 arc minutes.

Precision of the true node:

The true node can be computed from all of our three ephemerides.If you want a precision of the order of at
least one arc second, you have to choose either the JPL or the Swiss Ephemeris.

Maximum differences:

JPL-derived node – Swiss-Ephemeris-derived node~ 0.1 arc second

JPL-derived node – Moshier-derived node~
70arc seconds

(PLACALC was not better either. Its error was often > 1 arc minute.)

Distance of the true lunar node:

The distance of the true node is calculated on the basis of the
osculating ellipse of date.

The position of 'True Lilith' is given in the 'New International
Ephemerides' (NIE, Editions St. Michel) and in Francis Santoni 'Ephemerides de
la lune noire vraie 1910-2010' (Editions St. Michel, 1993). Both Ephemerides
coincide precisely.

The relation of this point to the mean apogee is not exactly of the same
kind as the relation between the true node and the mean node.Like the 'true' node, it can be considered as
an osculating orbital element of the lunar motion. But there is an important
difference: The apogee contains the concept of the ellipse, whereas the node
can be defined without thinking of an ellipse. As has been shown above, the
node can be derived from orbital planes or great circles, which is not possible
with the apogee. Now ellipses are good as a description of planetary orbits
because planetary orbits are close to a two-body problem. But they are not good
for the lunar orbit which is strongly perturbed by the gravity of the Sun
(three-body problem). The lunar orbit is far from being an ellipse!

The osculating apogee is 'true' twice a month: when it is in exact
conjunction with the Moon, the Moon is most distant from the earth; and when it
is in exact opposition to the moon, the moon is closest to the earth.The motion in between those two points, is an
oscillation with the period of a month. This oscillation is largely an artifact
caused by the reduction of the Moon’s orbit to a two-body problem. The
amplitude of the oscillation of the osculating apogee around the mean
apogee is +/- 30 degrees, while the true apogee's deviation from the
mean one never exceeds 5 degrees.

There is a small difference between the NIE's 'true Lilith' and our
osculating apogee, which results from an inaccuracy in NIE. The error reaches
20 arc minutes. According to Santoni, the point was calculated using 'les 58
premiers termes correctifs au perigée moyen' published by Chapront and
Chapront-Touzé. And he adds: ”Nous constatons que même en utilisant ces 58 termes correctifs,
l'erreur peut atteindre 0,5d!” (p. 13) We avoid this error, computing the orbital
elements from the position and the speed vectors of the moon. (By the way,
there is also an error of +/- 1 arc minute in NIE's true node. The reason is
probably the same.)

Precision:

The osculating apogee can be computed from any one of the three
ephemerides. If a precision of at least one arc second is required, one has to
choose either the JPL or the Swiss Ephemeris.

There have been several other attempts to solve the problem of a 'true'
apogee. They are not included in the SWISSEPH package.All of them work with a correction table.

They are listed in Santoni's 'Ephemerides de la lune noire vraie'
mentioned above. With all of them, a value is added to the mean apogee
depending on the angular distance of the sun from the mean apogee. There is
something to this idea. The actual apogees that take place once a month differ
from the mean apogee by never more than 5 degrees and seem to move along a
regular curve that is a function of the elongation of the mean apogee.

However, this curve does not have exactly the shape of a sine, as is
assumed by all of those correction tables.And most of them have an amplitude of more than 10 degrees, which is a
lot too high. The most realistic solution so far was the one proposed by Henry
Gouchon in ”Dictionnaire Astrologique”, Paris 1992, which is based on an
amplitude of 5 degrees.

In ”Meridian” 1/95, Dieter Koch has published another table that pays
regard to the fact that the motion does not precisely have the shape of a sine.
(Unfortunately, ”Meridian” confused the labels of the columns of the apogee and
the perigee.)

As has been said above, the osculating lunar apogee (so-called
"true Lilith") is a mathematical construct which assumes that the motion
of the moon is a two-body problem. This solution is obviously too simplistic.
Although Kepler ellipses are a good means to describe planetary orbits, they
fail with the orbit of the moon, which is strongly perturbed by the
gravitational pull of the sun. This solar perturbation results in gigantic
monthly oscillations in the ephemeris of the osculating apsides (the amplitude
is 30 degrees). These oscillations have to be considered an artifact of the insufficient model, they
do not really show a motion of the apsides.

A more sensible solution seems to be an interpolation between the real
passages of the moon through its apogees and perigees. It turns out that the
motions of the lunar perigee and apogee form curves of different quality and
the two points are usually not in opposition to each other. They are more or
less opposite points only at times when the sun is in conjunction with one of
them or at an angle of 90° from them. The amplitude of their oscillation about
the mean position is 5 degrees for the apogee and 25 degrees for the perigee.

This solution has been called the "interpolated"
or "realistic" apogee and perigee by Dieter Koch in his publications.
Juan Revilla prefers to call them the "natural"
apogee and perigee. Today, Dieter Koch would prefer the designation
"natural". The designation "interpolated" is a bit
misleading, because it associates something that astrologers used to do
everyday in old days, when they still used to work with printed ephemerides and
house tables.

Note on implementation (from Swiss Ephemeris Version 1.70 on):

Conventional interpolation algorithms do not work well in the case of
the lunar apsides. The supporting points are too far away from each other in
order to provide a good interpolation, the error estimation is greater than 1
degree for the perigee. Therefore, Dieter chose a different solution. He
derived an "interpolation method" from the analytical lunar theory
which we have in the form of moshier's lunar ephemeris. This
"interpolation method" has not only the advantage that it probably
makes more sense, but also that the curve and its derivation are both
continuous.

Differences between the Swiss
Ephemeris and other ephemerides of the osculation nodes and apsides are
probably due to different planetary ephemerides being used for their
calculation. Small differences in the planetary ephemerides lead to greater
differences in nodes and apsides.

Definitions of the nodes

Methods described in small
font are not supported by the Swiss Ephemeris software.

The lunar nodes are defined by
the intersection axis of the lunar orbital plane with the plane of the
ecliptic. At the lunar nodes, the moon crosses the plane of the ecliptic and
its ecliptic latitude changes sign. There are similar nodes for the planets,
but their definition is more complicated. Planetary nodes can be defined in the
following ways:

1)They can be understood as an axis defined by
the intersection line of two orbital planes. E.g., the nodes of Mars are
defined by the intersection line of the orbital plane of Mars with the plane of
the ecliptic (or the orbital plane of the Earth).

Note: However, as
Michael Erlewine points out in his elaborate web page on this topic
(http://thenewage.com/resources/articles/interface.html), planetary nodes could
be defined for any couple of planets. E.g. there is also an intersection line
for the two orbital planes of Mars and Saturn. Such non-ecliptic nodes have not
been implemented in the Swiss Ephemeris.

Because such lines
are, in principle, infinite, the heliocentric and the geocentric positions of
the planetary nodes will be the same. There are astrologers that use such
heliocentric planetary nodes in geocentric charts.

The ascending and
the descending node will, in this case, be in precise opposition.

2)There is a second definition that leads to different
geocentric ephemerides. The planetary nodes can be understood, not as an
infinite axis, but as the two points at which a planetary orbit
intersects with the ecliptic plane.

For the lunar nodes
and heliocentric planetary nodes, this definition makes no difference from the
definition 1). However, it does make a difference for geocentric
planetary nodes, where, the nodal points on the planets orbit are transformed
to the geocenter. The two points will not be in opposition anymore, or they
will roughly be so with the outer planets. The advantage of these nodes is that
when a planet is in conjunction with its node, then its ecliptic latitude will
be zero. This is not true when a planet is in geocentric conjunction with its
heliocentric node. (And neither is it always true for inner the planets, for
Mercury and Venus.)

Note: There is
another possibility, not implemented in the Swiss ephemeris: E.g., instead of
considering the points of the Mars orbit that are located in the ecliptic
plane, one might consider the points of the earth’s orbit that are
located in the orbital plane of Mars. If one takes these points geocentrically,
the ascending and the descending node will always form an approximate square.
This possibility has not been implemented in the Swiss Ephemeris.

3)Third, the planetary nodes could be defined as the intersection
points of the plane defined by their momentary geocentric position and motion
with the plane of the ecliptic. Here again, the ecliptic latitude would change
sign at the moment when the planet were in conjunction with one of its nodes.
This possibility has not been implemented in the Swiss Ephemeris.

Possible definitions for apsides and focal points

The lunar apsides - the lunar
apogee and lunar perigee - have already been discussed further above. Similar
points exist for the planets, as well, and they have been considered by
astrologers. Also, as with the lunar apsides, there is a similar disagreement:

One may consider either the
planetary apsides, i.e. the two points on a planetary orbitthat are closest to the sun and most distant
from the sun, resp. The former point is called the ”perihelion” and the
latter one the ”aphelion”. For a geocentric chart, these points could be
transformed from the heliocenter to the geocenter.

However, Bernard Fitzwalter
and Raymond Henry prefer to use the second focal points of the planetary
orbits. And they call them the ”black stars” or the ”black suns of the
planets”. The heliocentric positions of these points are identical to the
heliocentric positions of the aphelia, but geocentric positions are not identical,
because the focal points are much closer to the sun than the aphelia. Most of
them are even inside the Earth orbit.

The Swiss Ephemeris supports
both points of view.

Special case: the Earth

The Earth is a special case.
Instead of the motion of the Earth herself, the heliocentric motion of the
Earth-Moon-Barycenter (EMB) is used to determine the osculating perihelion.

There is no node of the earth
orbit itself.

There is an axis around which
the earth's orbital plane slowly rotates due to planetary precession. The
position points of this axis are not calculated by the Swiss Ephemeris.

Special case: the Sun

In addition to the Earth (EMB)
apsides, our software computes so-to-say "apsides" of the solar orbit
around the Earth, i.e. points on the orbit of the Sun where it is closest to
and where it is farthest from the Earth. These points form an opposition and
are used by some astrologers, e.g. by the Dutch astrologer George Bode or the
Swiss astrologer Liduina Schmed. The ”perigee”, located at about 13 Capricorn,
is called the "Black Sun", the other one, in Cancer, is called the
”Diamond”.

So, for a complete set of
apsides, one might want to calculate them for the Sun and the Earth and
all other planets.

Mean and osculating positions

There are serious problems about the ephemerides of planetary nodes and
apsides. There are mean ones and osculating ones. Both are well-defined points
in astronomy, but this does not necessarily mean that these definitions make
sense for astrology. Mean points, on the one hand, are not true, i.e. if a
planet is in precise conjunction with its mean node, this does not mean it be
crossing the ecliptic plane exactly that moment. Osculating points, on the
other hand, are based on the idealization of the planetary motions as two-body
problems, where the gravity of the sun and a single planet is considered and
all other influences neglected. There are no planetary nodes or apsides, at
least today, that really deserve the label ”true”.

Mean positions

Mean nodes and apsides
can be computed for the Moon, the Earth and the planets Mercury – Neptune. They
are taken from the planetary theory VSOP87. Mean points can not be
calculated for Pluto and the asteroids, because there is no planetary theory
for them.

Although the Nasa has
published mean elements for the planets Mercury – Pluto based on the JPL
ephemeris DE200, we do not use them (so far), because their validity is limited
to a 250 year period, because only linear rates are given, and because they are
not based on a planetary theory. (http://ssd.jpl.nasa.gov/elem_planets.html,
”mean orbit solutions from a 250 yr. least squares fit of the DE 200 planetary
ephemeris to a Keplerian orbit where each element is allowed to vary linearly
with time”)

The differences between the
DE200 and the VSOP87 mean elements are considerable, though:

NodePerihelion

Mercury 3”4”

Venus3”107”

Earth -35”

Mars74”4”

Jupiter330”1850”

Saturn178”1530”

Uranus806”6540”

Neptune225”11600” (>3 deg!)

Osculating nodes and apsides

Nodes and apsides can also be
derived from the osculating orbital elements of a body, the parameters that
define an ideal unperturbed elliptic (two-body) orbit for a given time.
Celestial bodies would follow such orbits if perturbations were to cease
suddenly or if there were only two bodies (the sun and the planet) involved in
the motion and the motion were an ideal ellipse. This ideal assumption
makes it obvious that it would be misleading to call such nodes or apsides
"true". It is more appropriate to call them "osculating".
Osculating nodes and apsides are "true" only at the precise moments,
when the body passes through them, but for the times in between, they are a
mere mathematical construct, nothing to do with the nature of an orbit.

We tried to solve the problem
by interpolating between actual passages of the planets through their
nodes and apsides. However, this method works only well with Mercury. With all
other planets, the supporting points are too far apart as to allow a sensible
interpolation.

There is another problem about
heliocentric ellipses. E.g. Neptune's orbit has often two perihelia and two
aphelia (i. e. minima and maxima in heliocentric distance) within one
revolution. As a result, there is a wild oscillation of the osculating or
"true" perihelion (and aphelion), which is not due to a
transformation of the orbital ellipse but rather due to the deviation of the
heliocentric orbit from an elliptic shape. Neptune’s orbit cannot be adequately
represented by a heliocentric ellipse.

In actuality, Neptune’s orbit
is not heliocentric at all. The double perihelia and aphelia are an effect of
the motion of the sun about the solar system barycenter. This motion is a lot
faster than the motion of Neptune, and Neptune cannot react to such fast
displacements of the Sun. As a result, Neptune seems to move around the
barycenter (or a mean sun) rather than around the real sun. In fact, Neptune's
orbit around the barycenter is therefore closer to an ellipse than his orbit
around the sun. The same is also true, though less obvious, for Saturn, Uranus
and Pluto, but not for Jupiter and the inner planets.

This fundamental problem about
osculating ellipses of planetary orbits does of course not only affect the
apsides but also the nodes.

As a solution, it seems reasonable
to compute the osculating elements of slow planets from their
barycentric motions rather than from their heliocentric motions. This procedure
makes sense especially for Neptune, but also for all planets beyond Jupiter. It
comes closer to the mean apsides and nodes for planets that have such points
defined. For Pluto and all trans-Saturnian asteroids, this solution may be used
as a substitute for "mean" nodes and apsides. Note, however, that
there are considerable differences between barycentric osculating and mean
nodes and apsides for Saturn, Uranus, and Neptune. (A few degrees! But
heliocentric ones are worse.)

Anyway, neither the
heliocentric nor the barycentric ellipse is a perfect representation of the
nature of a planetary orbit. So, astrologers should not expect anything very
reliable here either!

The best choice of method will
probably be:

For Mercury – Neptune: mean
nodes and apsides.

For asteroids that belong to
the inner asteroid belt: osculating nodes/apsides from a heliocentric ellipse.

For Pluto and transjovian
asteroids: osculating nodes/apsides from a barycentric ellipse.

The modes of the Swiss Ephemeris function
swe_nod_aps()

Thefunction swe_nod_aps() can be run in
the following modes:

1) Mean positions are given
for nodes and apsides of Sun, Moon, Earth, and the planets up to Neptune.
Osculating positions are given with Pluto and all asteroids. This is the
default mode.

2) Osculating positions are
returned for nodes and apsides of all planets.

3) Same as 2), but for planets
and asteroids beyond Jupiter, a barycentric ellipse is used.

4) Same as 1), but for Pluto
and asteroids beyond Jupiter, a barycentric ellipse is used.

For the reasons given above,
method 4) seems to make best sense.

In all of these modes, the second focal point of the ellipse can be
computed instead of the aphelion.

The standard distribution of SWISSEPH includes the main asteroids
Ceres, Pallas, Juno, Vesta, as well as 2060 Chiron, and 5145 Pholus. To compute
them, one must have the main-asteroid ephemeris files in the ephemeris
directory.

The names of these files are of the following form:

seas_18.se1main asteroids for 600 years from 1800 - 2400

The size of such a file is about 200 kb.

All other asteroids are available in separate files. The names of
additional asteroid files look like:

se00433.se1the
file of asteroid No. 433 (= Eros)

These files cover the period 3000 BC - 3000 AD.
A short version for the years 1500 – 2100 AD has the file name with an 's'
imbedded, se00433s.se1.

The numerical integration of the all numbered asteroids is an ongoing
effort. In December 1998, 8000 asteroids were numbered, and their orbits
computed by the devlopers of Swiss Ephemeris. In January 2001, the list of
numbered asteroids reached 20957, in January 2014 more than 380’000, and it is
still growing very fast.

Any asteroid can be called either with the JPL, the Swiss, or the
Moshier ephemeris flag, and the results will be slightly different. The reason
is that the solar position (which is needed for geocentric positions) will be
taken from the ephemeris that has been specified.

Availability of asteroid files:

-all short files
(over 200000) are available for free download at our ftp server ftp.astro.ch/pub/swisseph.
The purpose of providing this large number of files for download is that the
user can pick those few asteroids he/she is interested in.

-for all named
asteroids also a long(6000 years) file
is available in the download area.

To generate our asteroid ephemerides, we have modified the numerical
integrator of Steve Moshier, which was capable to rebuild the DE200 JPL
ephemeris.

Orbital elements, with a few exceptions, were taken from the asteroid
database computed by E. Bowell, Lowell Observatory, Flagstaff, Arizona
(astorb.dat). After the introduction of the JPL database mpcorb.dat, we still
keep working with the Lowell data because Lowell elements are given with one
more digit, which can be relevant for long-term integrations.

For a few close-Sun-approaching asteroids like 1566 Icarus, we use the
elements of JPL’s DASTCOM database. Here, the Bowell elements are not good for
long term integration because they do not account for relativity.

Our asteroid ephemerides take into account the gravitational
perturbations of all planets, including the major asteroids Ceres, Pallas, and
Vesta and also the Moon.

The mutual perturbations of Ceres, Pallas, and Vesta were included by
iterative integration. The first run was done without mutual perturbations, the
second one with the perturbing forces from the positions computed in the first
run.

The precision of our integrator is very high. A test integration of the
orbit of Mars with start date 2000 has shown a difference of only 0.0007 arc
second from DE200 for the year 1600. We also compared our asteroid ephemerides
with data from JPL’s on-line ephemeris system ”Horizons” which provides
asteroid positions from 1600 on. Taking into account that Horizons does not
consider the mutual perturbations of the major asteroids Ceres, Pallas and
Vesta, the difference is never greater than a few 0.1 arcsec.

(However, the Swisseph asteroid ephemerides do consider those
perturbations, which makes a difference of 10 arcsec for Ceres and 80 arcsec
for Pallas. This means that our asteroid ephemerides are even better than the
ones that JPL offers on the web.)

The accuracy limits are therefore not set by the algorithms of our
program but by the inherent uncertainties in the orbital elements of the
asteroids from which our integrator has to start.

Sources of errors are:

-Only some of the
minor planets are known to better than an arc second for recent decades. (See
also informations below on Ceres, Chiron, and Pholus.)

-Bowells elements do
not consider relativistic effects, which leads to significant errors with
long-term integrations of a few close-Sun-approaching orbits (except 1566,
2212, 3200, 5786, and 16960, for which we use JPL elements that do take into
account relativity).

The orbits of some asteroids are extremely sensitive to perturbations by
major planets. E.g. 1862 Apollo becomes chaotic before the year 1870 AD when he
passes Venus within a distance which is only one and a half the distance from
the Moon to the Earth. In this moment, the small uncertainty of the initial
elements provided by the Bowell database grows, so to speak, ”into infinity”,
so that it is impossible to determine the precise orbit prior to that date. Our
integrator is able to detect such happenings and end the ephemeris generation
to prevent our users working with meaningless data.

The orbital elements of the four main asteroids Ceres, Pallas, Juno, and
Vesta are known very precisely, because these planets have been discovered
almost 200 years ago and observed very often since. On the other hand, their
orbits are not as well-determined as the ones of the main planets. We estimate
that the precision of the main asteroid ephemerides is better than 1 arc second
for the whole 20th century. The deviations from the Astronomical Almanac
positions can reach 0.5” (AA 1985 – 1997). But the tables in AA are based on
older computations, whereas we used recent orbital elements. (s. AA 1997, page L14)

MPC elements have a precision of five digits with mean anomaly,
perihelion, node, and inclination and seven digits with eccentricity and
semi-axis. For the four main asteroids, this implies an uncertainty of a few
arc seconds in 1600 AD and a few arc minutes in 3000 BC.

Positions of Chiron can be well computed for the time between 700
ADand 4650 AD. As a result of close
encounters with Saturn in Sept. 720 AD and in 4606 AD we cannot trace its orbit
beyond this time range. Small uncertainties in today's orbital elements have chaotic
effects before the year 700.

Do not rely on earlier Chiron ephemerides supplying a Chiron for
Cesar's, Jesus', or Buddha's birth chart. They are completely meaningless.

Pholus is a minor planet with orbital characteristics that are similar
to Chiron's. It was discovered in 1992. Pholus' orbital elements are not yet as
well-established as Chiron's. Our ephemeris is reliable from 1500 AD through
now. Outside the 20th century it will probably have to be corrected by several
arc minutes during the coming years.

Dieter Koch has written the application program Ceres which
allows to compute all kinds of lists for asteroid astrology. E.g. you can
generate a list of all your natal asteroids ordered by position in the zodiac.
But the program does much more:

We include some astrological factors in the ephemeris which have no
astronomical basis – they have never been observed physically. As the purpose
of the Swiss Ephemeris is astrology, we decided to drop our scientific view in
this area and to be of service to those astrologers who use these
‘hypothetical’ planets and factors. Of course neither of our scientific
sources, JPL or Steve Moshier, have anything to do with this part of the Swiss
Ephemeris.

There have been discussions whether these factors are to be called
'planets' or 'Transneptunian points'. However, their inventors, the German
astrologers Witte and Sieggrün, considered them to be planets. And moreover
they behave like planets in as far as they circle around the sun and obey its
gravity.

On the other hand, if one looks at their orbital elements, it is obvious
that these orbits are highly unrealistic.Some of them are perfect circles – something that does not exist in
physical reality. The inclination of the orbits is zero, which is very
improbable as well. The revised elements published by James Neely in Matrix
Journal VII (1980) show small eccentricities for the four Witte planets, but
they are still smaller than the eccentricity of Venus which has an almost
circular orbit. This is again very improbable.

There are even more problems. An ephemeris computed with such elements
describes an unperturbed motion, i.e. it takes into account only the Sun's
gravity, not the gravitational influences of the other planets. This may result
in an error of a degree within the 20thcentury, and greater errors for earlier
centuries.

Also, note that none of the real transneptunian objects that have been
discovered since 1992 can be identified with any of the Uranian planets.

SWISSEPH uses James Neely's revised orbital elements, because they agree
better with the original position tables of Witte and Sieggrün.

The hypothetical planets can again be called with any of the three
ephemeris flags. The solar position needed for geocentric positions will then
be taken from the ephemeris specified.

This hypothetical planet was postulated 1946 by the French astronomer
M.E. Sevin because of otherwise unexplainable gravitational perturbations in
the orbits of Uranus and Neptune.

However, this theory has been superseded by other attempts during the
following decades, which proceeded from better observational data.They resulted in bodies and orbits completely
different from what astrologers know as 'Isis-Transpluto'. More recent studies
have shown that the perturbation residuals in the orbits of Uranus and Neptune
are too small to allow postulation of a new planet. They can, to a great
extent, be explained by observational errors or by systematic errors in sky
maps.

In telescope observations, no hint could be discovered that this planet
actually existed. Rumors that claim the opposite are wrong.Moreover, all of the transneptunian bodies
that have been discovered since 1992 are very different from Isis-Transpluto.

Even if Sevin's computation were correct, it could only provide a rough
position. To rely on arc minutes would be illusory.Neptune was more than a degree away from its
theoretical position predicted by Leverrier and Adams.

Moreover, Transpluto's position is computed from a simple Kepler
ellipse, disregarding the perturbations by other planets' gravities.Moreover, Sevin gives no orbital inclination.

Though Sevin gives no inclination for his Transpluto, you will realize
that there is a small ecliptic latitude in positions computed by SWISSEPH. This
mainly results from the fact that its orbital elements are referred to epoch
5.10.1772 whereas the ecliptic changes position with time.

The elements used by SWISSEPH are taken from ”Die Sterne” 3/1952, p. 70.
The article does not say which equinox they are referred to.Therefore, we fitted it to the Astron
ephemeris which apparently uses the equinox of 1945 (which, however, is rather
unusual!).

This is another attempt to predict Planet X's orbit and position from
perturbations in the orbits ofUranus
and Neptune. It was published in The Astronomical Journal 96(4), October 1988,
p. 1476ff. Its precision is meant to be of the order of +/- 30 degrees.
According to Harrington there is also the possibility that it is actually
located in the opposite constellation, i.e. Taurus instead of Scorpio. The
planet has a mean solar distance of about 100 AU and a period of about 1000
years.

A highly speculative planet derived from the theory of Zecharia Sitchin,
who is an expert in ancient Mesopotamian history and a ”paleoastronomer”.The elements have been supplied by Christian
Woeltge, Hannover.This planet is
interesting because of its bizarre orbit. It moves in clockwise direction and
has a period of 3600 years. Its orbit is extremely eccentric. It has its
perihelion within the asteroid belt, whereas its aphelion lies at about 12
times the mean distance of Pluto.In
spite of its retrograde motion, it seems to move counterclockwise in
recent centuries. The reason is that it is so slow that it does not even
compensate the precession of the equinoxes.

This is a ‘hypothetical’ planet inside the orbit of Mercury (not
identical to the “Uranian” planet Vulkanus). Orbital elements according to L.H.
Weston. Note that the speed of this “planet” does not agree with the Kepler
laws. It is too fast by 10 degrees per year.

This is a ‘hypothetical’ second moon of the earth (or a third one, after
the “Black Moon”) of obscure provenance. Many Russian astrologers use it. Its
distance from the earth is more than 20 times the distance of the moon and it
moves about the earth in 7 years. Its orbit is a perfect, unperturbed circle.
Of course, the physical existence of such a body is not possible. The gravities
of Sun, Earth, and Moon would strongly influence its orbit.

This is another hypothetical second moon of the earth, postulated by a
Dr. Waldemath in the Monthly Wheather Review 1/1898. Its distance from
the earth is 2.67 times the distance of the moon, its daily motion about 3
degrees. The orbital elements have been derived from Waldemath’s original data.
There are significant differences from elements used in earlier versions of
Solar Fire, due to different interpretations of the values given by Waldemath.
After a discussion between Graham Dawson and Dieter Koch it has been agreed
that the new solution is more likely to be correct. The new ephemeris does not
agree with Delphine Jay’s ephemeris either, which is obviously inconsistent
with Waldemath’s data.

This body has never been confirmed. With its 700-km diameter and an
apparent diameter of 2.5 arc min, this should have been possible very soon
after Waldemath’s publication.

These are the hypothetical planets that have lead to the discovery of
Neptune and Pluto or at least have been brought into connection with them.Their enormous deviations from true Neptune
and Pluto may be interesting for astrologers who work with hypothetical bodies.
E.g. Leverrier and Adams are good only around the 1840ies, the discovery epoch
of Neptune. To check this, call the program swetest as follows:

$ swetest -p8 -dU -b1.1.1770 -n8 -s7305 -hel
-fPTLBR -head

(i.e.: compute planet 8 (Neptune) - planet 'U' (Leverrier), from
1.1.1770, 8 times, in 7305-day-steps, heliocentrically. You can do this from
the Internet web page swetest.htm. The
output will be:)

Nep-Lev
01.01.1770-18° 0'52.38110°55' 0.0332-6.610753489

Nep-Lev
01.01.1790-8°42' 9.11131°42'55.7192-4.257690148

Nep-Lev
02.01.1810-3°49'45.20141°35'12.0858-2.488363869

Nep-Lev
02.01.1830-1°38' 2.80760°35'57.0580-2.112570665

Nep-Lev
02.01.18501°44'23.0943-0°43'38.5357-3.340858070

Nep-Lev
02.01.18709°17'34.4981-1°39'24.1004-5.513270186

Nep-Lev
02.01.189021°20'56.6250-1°38'43.1479-7.720578177

Nep-Lev
03.01.191036°27'56.1314-0°41'59.4866-9.265417529

(difference in(difference in(difference in

longitude)latitude)solar distance)

One can see that the error is in the range of 2 degrees between 1830 and
1850 and grows very fast beyond that period.

One of the
main differences between the western and the eastern tradition of astrology is
the definition of the zodiac. Western astrology uses the so-called tropical
zodiac in which 0 Aries is defined by the vernal point (the celestial point
where the sun stands at the beginning of spring). The tropical zodiac is a division of the ecliptic into 12 equal-sized zodiac signs of 30° each. Astrologers call these signs after
constellations that are found along the ecliptic, although they are actually
independent of these constellations. Due to the precession of the equinox, the
vernal point and tropical Aries move through all constellations along the
ecliptic, staying for roughly 2160 years in each one of them. Currently, the
beginning of tropical Aries is located in the constellation of Pisces. In a few
hundred years, it will enter Aquarius, which is the reason why the more
impatient ones among us are already preparing for the “Age of Aquarius”.

There are
also sidereal traditions of astrology, both a Hindu tradition and a western
tradition, which derives itself from ancient Hellenistic and Babylonian
astrology. They use a so-called sidereal zodiac, which consists of 12
equal-sized zodiac signs, too, but it is tied to some fixed reference point,
i.e. usually some fixed star. These sidereal zodiac signs only roughly coincide
with the sidereal zodiacal constellations, which are of variable size.

While the
definition of the tropical zodiac is very obvious and never questioned,
sidereal astrology has considerable problems in defining its zodiac. There are
many different definitions of the sidereal zodiac that differ by several
degrees from each other. At first glance, all of them look arbitrary, and there
is no striking evidence – from a mere astronomical point of view – for anyone
of them. However, a historical study shows at least that many of them are
related to each other and the basic approaches aren’t so many.

Sidereal
planetary positions are usually computed from tropicl positions using the equation:

sidereal_position = tropical_position –
ayanamsha(t) ,

where ayanamsha
is the difference between the two zodiacs at a given epoch. (Sanskrit ayanâmsha
means ”part of a solar path (or half year)”; the Hindi form of the word is ayanamsa
with an s instead of sh.)

The value
of the ayanamsha of date is usually computed from the ayanamsha
value at a particular start date (e.g. 1 Jan 1900) and the speed of the vernal
point, the so-called precession rate in ecliptic longitude.

The zero
point of the sidereal zodiac is therefore traditionally defined by the
equation:

sidereal_Aries = tropical Aries +
ayanamsha(t).

The Swiss
Ephemeris offers about fourty different ayanamshas, but the user can
also define his or her own ayanamsha.

There have
been several attempts to calculate the zero point of the Babylonian ecliptic
from cuneiform lunar and planetary tablets. Positions were given relative to
some sidereally fixed reference point. The main problem in fixing the zero
point is the inaccuracy of ancient observations. Around 1900 F.X. Kugler found
that the Babylonian star positions fell into three groups:

Kugler
ayanamshas:

9) ayanamsha = -3°22´, t0 = -100

10) ayanamsha
= -4°46´, t0 = -100Spica at 29 vi
26

11) ayanamsha
= -5°37´, t0 = -100

(9 – 11 =
Swiss Ephemeris ayanamsha numbers)

In 1958, Peter
Huber reviewed the topic in the light of new material and found:

12) Huber
Ayanamsha:

ayanamsha = -4°28´ +/- 20´, t0 = –100Spica at 29 vi 07’59”

The
standard deviation was 1°08’

(Note, this
ayanamsha was corrected with SE
version 2.05. A wrong value of -4°34’ had been taken over from Mercier, “Studies
on the Transmission of Medieval Mathematical Astronomy”, IIb, p. 49.)

In 1977 Raymond
Mercier noted that the zero point might have been defined as the ecliptic
point that culminated simultaneously with the star eta Piscium (Al Pherg).
For this possibility, we compute:

13) Eta
Piscium ayanamsha:

ayanamsha = -5°04’46”, t0 = –129Spica at 29 vi 21

Around
1950, Cyril Fagan, the founder of the modern western sidereal astrology,
reintroduced the old Babylonian zodiac into astrology, placing the fixed star
Spica near 29°00 Virgo. As a result of “rigorous statistical investigation”
(astrological!) of solar and lunar ingress charts, Donald Bradley decided
that the sidereal longitude of the vernal point must be computed from Spica at
29 vi 06'05" disregarding its proper motion. Fagan and Bradley
defined their ”synetic vernal point” as:

The
difference between P. Huber’s zodiac and the Fagan/Bradley ayanamsha is smaller
than 1’.

According
to a text by Fagan (found on the internet), Bradley ”once opined in print prior
to "New Tool" that it made more sense to consider Aldebaran and
Antares, at 15 degrees of their respective signs, as prime fiducials than it
did to use Spica at 29 Virgo”. Such statements raise the question if the
sidereal zodiac ought to be tied up to one of those stars.

For this
possibility, Swiss Ephemeris gives an Aldebaran ayanamsha:

14) Aldebaran-Antares
ayanamsha:

ayanamsha with Aldebaran at 15ta00’00” and
Antares at 15sc00’17” around the year –100.

The
difference between this ayanamsha and the Fagan/Bradley one is 1’06”.

In 2010,
the astronomy historian John P. Britton made another investigation in cuneiform
astronomical tablets and corrected Huber’s by a 7 arc minutes.

Raymond
Mercier has shown
that all of the ancient Greek and the medieval Arabic astronomical works
located the zero point of the ecliptic somewhere between 10 and 22 arc
minutes east of the star zeta Piscium. He is of the opinion that this
definition goes back to the great Greek astronomer Hipparchus.

Mercier
points out that according to Hipparchus’ star catalogue, the stars alpha
Arietis, beta Arietis, zeta Piscium, and Spica are in a very precise
alignment on a great circle which goes through that zero point near zeta
Piscium. Moreover, this great circle was identical with the horizon once a
day at Hipparchus’ geographical latitude of 36°. In other words, the zero point
rose at the same time when the three mentioned stars in Aries and Pisces rose
and when Spica set.

This would
of course be a nice definition for the zero point, but unfortunately the stars
were not really in such precise alignment. They were only assumed to be
so.

Mercier
gives the following ayanamshas for Hipparchus and Ptolemy
(who used the same star catalogue as Hipparchus):

(According
to Mercier’s calculations, the Hipparchan zero point should have been between
12 and 22 arc min east of zePsc, but the Hipparchan ayanamsha, as given
by Mercier, has actually the zero point 26’ east of zePsc. This comes from the
fact that Mercier refers to the Hipparchan position of zeta Piscium,
which was at least rounded to 10’, if correct at all.)

Using the
information that Aries rose when Spica set at a geographical latitude of
36 degrees, the precise ayanamsha would be -8°58’13” for 27 June –128
(jd 1674484) and zePsc would be found at 29pi12’, which is too far from the
place where it ought to be.

Mercier
also discusses the old Indian precession models and zodiac point definitions.
He notes that, in the Sûryasiddhânta, the star zeta Piscium (in
Sanskrit Revatî) has almost the same position as in the Greek sidereal
zodiac, i.e. 29°50’ in Pisces. On the other hand, however, Spica (in Sanskrit Citrâ)
is given the longitude 30° Virgo. Unfortunately, these positions of Revatî and Citrâ/Spica are incompatible; either Spica or Revatî must be
considered wrong.

Moreover,
if the precession model of the Sûryasiddânta is used to compute an ayanamsha
for the date of Hipparchus, it will turn out to be –9°14’01”, which is very
close to the Hipparchan value. The same calculation can be done with the Âryasiddânta,
and the ayanamsha for Hipparchos’ date will be –9°14’55”. For the Siddânta
Shiromani the zero point turns out to be Revatî itself. By the way, this is
also the zero point chosen by Copernicus! So, there is an astonishing
agreement between Indian and Western traditions!

The same
zero point near the star Revatî is also used by the so-called Ushâ-Shashî
ayanamsha. It differs from the Hipparchan one by only 11 arc minutes.

4) Usha-Shashi ayanamsha:

ayanamsha = 18°39’39.461 Jan. 1900

zePsc (Revatî) 29pi50’ (today),
29pi45’ (Hipparchus’ epoch)

The
Greek-Arabic-Hindu ayanamsha was zero around 560 AD. The tropical and
the sidereal zero points were at exactly the same place.

In the year
556, under the Sassanian ruler Khusrau Anûshirwân, the astronomers of Persia
met to correct their astronomical tables, the so-called Zîj al-Shâh.
These tables are no longer extant, but they were the basis of later Arabic
tables, the ones of al-Khwârizmî and the Toledan tables.

One of the
most important cycles in Persian astronomy/astrology was the synodic cycle of
Jupiter, which started and ended with the conjunctions of Jupiter with the Sun.
This cycle happened to end in the year 564, and the conjunction of
Jupiter with the Sun took place only one day after the spring equinox. And the
spring equinox took place precisely 10 arcmin east of zePsc. This may be a
mere coincidence from a present-day astronomical point of view, but for
scientists of those days this was obviously the moment to redefine all
astronomical data.

Mercier
also shows that in the precession (trepidation) model used in that time and in
other models used later by Arabic astronomers, precession was considered to be
a phenomenon connected with “the movement of Jupiter, the calendar marker of
the night sky, in its relation to the Sun, the time keeper of the daily sky”.
Such theories were of course wrong, from the point of view of modern knowledge,
but they show how important that date was considered to be.

After the
Sassanian reform of astronomical tables, we have a new definition of the
Greek-Arabic-Hindu sidereal zodiac (this is not explicitly stated by Mercier,
however):

16) Sassanian
ayanamsha:

ayanamsha = 018
Mar 564, 7:53:23 UT (jd /ET 1927135.8747793)Sassanian

zePsc29pi49'59"

The same
zero point then reappears with a precision of 1’ in the Toledan tables, the
Khwârizmian tables, the Sûrya Siddhânta, and the Ushâ-Shashî ayanamsha.

Sources:

- Raymond
Mercier, ”Studies in the Medieval Conception of Precession”,

The explanations above are mainly
derived from the article by Mercier. However, it is possible to derive
ayanamshas from ancient Indian works themselves.

The planetary theory of the main
work of ancient Indian astronomy, the Suryasiddhanta, uses the so-called
Kaliyuga era as its zero point, i. e. the 18th February 3102 BC,
0:00 local time at Ujjain, which is at geographic longitude of 75.7684565 east
(Mahakala temple). This era is henceforth called “K0s”. This is also the zero
date for the planetary theory of the ancient Indian astronomer Aryabhata, with
the only difference that he reckons from sunrise of the same date instead of
midnight. We call this Aryabhatan Kaliyuga era “K0a”.

Aryabhata mentioned that he was 23
years old when exactly 3600 years had elapsed since the beginning of the
Kaliyuga era. If 3600 years with a year length as defined by the Aryabhata are
counted from K0a, we arrive at the 21st March, 499 AD, 6:56:55.57
UT. At this point of time the mean Sun is assumed to have returned to the
beginning of the sidereal zodiac, and we can try to derive an ayanamsha from
this information. There are two possible solutions, though:

1. We can find the place of the mean
Sun at that time using modern astronomical algorithms and define this point as
the beginning of the sidereal zodiac.

2. Since Aryabhata believed that the
zodiac began at the vernal point, we can take the vernal point of this date as
the zero point.

The same calculations can be done
based on K0s and the year length of the Suryasiddhanta. The resulting date of
Kali 3600 is the same day but about half an hour later: 7:30:31.57 UT.

Algorithms for the mean Sun were
taken from: Simon et alii, “Numerical expressions for precession formulae and
mean elements for the Moon and the planets”, in: Astron. Astrophys. 282,663-683
(1994).

According
to Govindasvamin (850 n. Chr.), Aryabhata and his disciples taught that the
vernal point was at the beginning of sidereal Aries in the year 522 AD (= Shaka
444). This tradition probably goes back to an erroneous interpretation of
Aryabhata's above-mentioned statement that he was 23 years old when 3600 had elapsed
after the beginning of the Kaliyuga. For the sake of completeness, we therefore
add the following ayanamsha:

There is
another ayanamsha tradition that assumes the star Spica (in Sanskrit Citra) at
0° Libra. This ayanamsha is the most common one in modern Hindu astrology. It
was first proposed by the astronomy historian S. B. Dixit (also written
Dikshit), who in 1896 published his important work History of Indian
Astronomy (= Bharatiya Jyotih
Shastra; bibliographical details further below). Dixit came to the
conclusion that, given the prominence that Vedic religion gave to the cardinal
points of the tropical year, the Indian calendar, which is based on the zodiac,
should be reformed and no longer be calculated relative to the sidereal, but to
the tropical zodiac. However, if such a reform could not be brought about due
to the rigid conservatism of contemporary Vedic culture, then the ayanamsha
should be chosen in such a way that the sidereal zero point would be in
opposition to Spica. In this way, it would be in accordance with Grahalaghava, a work by the 16th century
astronomer Ganeśa Daivajña that
was still used in the 20th century by Indian calendar makers. (op.
cit., Part II, p. 323ff.). This view was taken over by the Indian Calendar Reform Committee on the occasion of the Indian
calendar reform in 1956, when the ayanamsha based on the star
Spica/Citra was declared the Indian standard. This standard is mandatory not
only for astrology but also for astronomical ephemerides and almanacs and
calendars published in India. The ayanamsha based on the star Spica/Citra
became known as “Lahiri ayanamsha”. It was named after the Calcuttan astronomer
and astrologer Nirmala Chandra Lahiri, who was a member of the Reform
Committee.

However, as
has been said, it was Dixit who first propagated this solution to the ayanamsha
problem. In addition, the Suryasiddhanta, the most important work of ancient
Hindu astronomy, which was written in the first centuries AD, but reworked
several times, already assumes Spica/Citra at 180° (although this statement has
caused a lot of controversy because it is in contradiction with the positions
of other stars, in particular with zeta Piscium/Revati at 359°50‘). Finally yet
importantly, the same ayanamsha seems to have existed in Babylon and Greece, as
well. While the information given above in the chapters about the Babylonian
and the Hipparchan traditions are based on analyses of old star catalogues and
planetary theories, a study by Nick Kollerstrom of 22 ancient Greek and 5
Babylonian birth charts seems to prove that they fit better with Spica at 0
Libra (= Lahiri), than with Aldebaran at 15 Taurus and Spica at 29 Virgo (=
Fagan/Bradley).

The
standard definition of the Indian ayanamsha (“Lahiri” ayanamsha) was originally
introduced in 1955 by the Indian Calendar
Reform Committee (23°15' 00" on the 21 March 1956, 0:00 Ephemeris
Time). The definition was corrected in Indian
Astronomical Ephemeris 1989, page 556, footnote:

"According
to new determination of the location of equinox this initial value has been
revised to and used in computing the mean ayanamsha with effect from
1985'."

The mention
of “mean ayanamsha” is misleading though. The value 23°15' 00".658 is true
ayanamsha, i. e. it includes nutation and is relative to the true equinox of
date.

The Lahiri
standard position of Spica is 179°59’04 in the year 2000, and 179°59’08 in
1900. In the year 285, when the star was conjunct the autumnal equinox, its
position was 180°00’16. It was only in the year 667 AD that its position was exactly
180°. The motion of the star is partly caused by its proper motion and partly by
the so-called planetary precession, which causes very slow changes in the orientation
of the ecliptic plane. But what method exactly was used to define this
ayanamsha? According to the Indian pundit A.K. Kaul, an expert in Hindu
calendar and astrology, Lahiri wanted to place the star at 180°, but at the
same time arrive at an ayanamsha that was in agreement with the Grahalaghava,
an important work for traditional Hindu calendar calculation that was written
in the 16th century. (e-mail from Mr. Kaul to Dieter Koch on 1 March
2013)

Swiss
Ephemeris versions below 1.78.01, had a slightly different definition of the
Lahiri ayanamsha that had been taken from Robert Hand's astrological software
Nova. The correction amounts to 0.01 arc sec.

In 1967, 12
years after the standard definition of the Lahiri ayanamsha had been published
by the Calendar Reform Committee, Lahiri published another Citra ayanamsha in
his Bengali book Panchanga Darpan.
There, the value of “mean ayanamsha” is given as 22°26’45”.50 in 1900, whereas
the official value is 22°27’37”.76. The intention behind this modification is
obvious. With the standard Lahiri ayanamsha, the position of Spica was “wrong”,
i.e. it deviated from 180° by almost an arc minute. Lahir obviously wanted to
place the star exactly at 180° for recent years. It therefore seems that Lahiri
did not follow the Indian standard himself but was of the opinion that Spica
had to be at exactly 180° (true chitrapaksha ayanamsha). The Swiss Ephemeris
does not support this updated Lahiri ayanamsha. Users who want to follow
Lahiri’s real intention are advised to use the True Chitrapaksha ayanamsha (No.
27, see below).

Many thanks
to Vinay Jha, Narasimha Rao, and Avtar Krishen Kaul for their help in our
attempt to understand this complicated matter.

Additional Citra/Spica
ayanamshas:

The
Suryasiddhanta gives the position of Spica/Citra as 180° in polar longitude
(ecliptic longitude, but projection on meridian lines). From this, the
following Ayanamsha can be derived:

Usually
ayanamshas are defined by an epoch and an initial ayanamsha offset. However, if
one wants to make sure that a particular fixed star always remains at a precise
position, e. g. Spica at 180°, it does not work this way. The correct procedure
for this to work is to calculate the tropical position of Spica for the date
and subtract it from the tropical position of the planet:

The
Suryasiddhanta also mentions that Revati/zeta-Piscium is exactly at 359°50’ in
polar ecliptic longitude (projection onto the ecliptic along meridians).
Therefore the following two ayanamshas were added:

(Note, this
was incorrectly implemented in SE 2.00 – SE 2.04. The Position of Revati was
0°. Only from SE 2.05 on, this ayanamsha is correct.)

Siddhantas
usually assume the star Pushya (delta Cancri = Asellus Australis) at 106°. PVR
Narasimha Rao believes this star to be the best anchor point for a sidereal
zodiac. In the Kalapurusha theory, which assigns zodiac signs to parts of the
human body, the sign of Cancer is assigned to the heart, and according to Vedic
spiritual literature, the root of human existence is in the heart. Mr.
Narasimha Rao therefore proposed the following ayanamsha:

29) True
Pushya paksha ayanamsha

Pushya/deCnC
is always exactly at longitude 106°.

Another
ayanamsha close to the Lahiri ayanamsha is named after the Indian astrologer K.S.
Krishnamurti (1908-1972).

The
definition of the tropical zodiac is very simple and convincing. It starts at
one of the two intersection points of the ecliptic and the celestial equator.
Similarly, the definition for the house circle which is said to be an analogy
of the zodiac, is very simple. It starts at one of the two intersection points
of the ecliptic and the local horizon. Unfortunately, sidereal traditions do
not provide such a simple definition for the sidereal zodiac. The sidereal
zodiac is always fixed at some anchor star such as Citra (Spica), Revati (zeta
Piscium), or Aldebaran and Antares.

Unfortunately,
nobody can tell why any of these stars should be so important that it could be
used as an anchor point for the zodiac. In addition, all these solutions are
unattractive in that the fixed stars actually are not fixed

forever,
but have a small proper motion which over a long period of time such as several
millennia, can result in a considerable change in position. While it is
possible to tie the zodiac to the star Spica in a way that it remains at 0° Libra
for all times, all other stars would change their positions relative to Spica
and relative to this zodiac and would not be fixed at all. The appearance of
the sky changes over long periods of time. In 100’000 years, the

constellation
will look very different from now, and the nakshatras (lunar mansions) will get
confused. For this reason, a zodiac defined by positions of stars is
unfortunately not able to provide an everlasting reference frame.

For such or
also other reasons, some astrologers (Raymond Mardyks, Ernst Wilhelm, Rafael
Gil Brand, Nick Anthony Fiorenza) have tried to define the sidereal zodiac
using either the galactic centre or the node of the galactic equator with the
ecliptic. It is obvious that this kind of solution, which would not depend on
the position of a single star anymore, could provide a philosophically
meaningful and very stable definition of the zodiac. Fixed stars would be
allowed to change their positions over very long periods of time, but the
zodiac could still be considered fixed and “sidereal”.

The Swiss
astrologer Bruno Huber has pointed out that everytime the Galactic Center enters
the next tropical sign the vernal point enters the previous sidereal sign. E.g.,
around the time the vernal point will enter Aquarius (at the beginning of the
so-called Age of Aquarius), the Galactic Center will enter from Sagittarius into
Capricorn. Huber also notes that the ruler of the tropical sign of the Galactic
Center is always the same as the ruler of the sidereal sign of the vernal point
(at the moment Jupiter, will be Saturn in a few hundred years).

A
correction of the Fagan ayanamsha by about 2 degrees or a correction of
the Lahiri ayanamsha by 3 degrees would place the Galactic Center at 0
Sagittarius. Astrologically, this would obviously make some sense. Therefore,
we added an ayanamsha fixed at the Galactic Center in 1999 in Swiss
Ephemeris 1.50, when we introduced sidereal ephemerides (suggestion by D. Koch,
without any astrological background):

A different
solution was proposed by the American astrologer Ernst Wilhelm in 2004. He
projects the galactic centre on the ecliptic in polar projection, i.e. along a
great circle that passes through the celestial north pole (in Sanskrit dhruva) and the galactic centre. The
point at which this great circle cuts the ecliptic is defined as the middle of
the nakshatra Mula, which corresponds to sidereal 6°40’ Sagittarius.

36)
Dhruva Galactic Center Middle Mula Ayanamsha (Ernst Wilhelm)

For Hindu
astrologers who follow a tradition oriented towards the star Revati (ζ
Piscium), this solution may be particularly interesting because when the
galactic centre is in the middle of Mula, then Revati is almost exactly at the
position it has in Suryasiddhanta, namely 29°50 Pisces. Also interesting in
this context is the fact that the meaning of the Sanskrit word mūlam is “root, origin”. Mula may
have been the first of the 27 nakshatras in very ancient times, before the
Vedic nakshatra circle and the Hellenistic zodiac were conflated and Ashvini,
which begins at 0° Aries, became the first nakshatra.

Another
ayanamsha based on the galactic centre was proposed by the German-Spanish
astrologer Rafael Gil Brand. Gil Brand places the galactic centre at the golden
section between 0° Scorpion and 0° Aquarius. The

axis
between 0° Leo and 0° Aquarius is the axis of the astrological ruler system.

The other
possibility – in analogy with the tropical ecliptic and the house circle –
would be to start the sidereal ecliptic at the intersection point of the
ecliptic and the galactic plane. At present, this point is located near 0
Capricorn. However, defining this point as sidereal 0 Aries would mean to break
completely with the tradition, because it is far away from the traditional
sidereal zero points.

The sidereal zodiac
and the Galactic Equator

Other ways
to define a sidereal zodiac without any dependence of anchor stars use the
intersection point (or node) of the galactic equator with the ecliptic. This point
has been used with the following ayanamshas:

34)
Skydram Ayanamsha (Raymond Mardyks)

(also known
as Galactic Alignment Ayanamsha)

This
ayanamsha was proposed in 1991 by the American astrologer Raymond Mardyks. It
had the value 30° on the autumn equinox 1998. Consequently, the node (intersection
point) of the galactic equator with the ecliptic was very close to sidereal 0°
Sagittarius on the same date, and there was an interesting “cosmic alignment”:
The galactic pole pointed exactly towards the autumnal equinoctial point, and
the galactic-ecliptic node coincided with the winter solstitial point (tropical
0° Capricorn).

Mardyks'
calculation is based on the galactic coordinate system that was defined by the
International Astronomical Union in 1958.

This is a
variation of Mardyks' Skydram or "Galactic Alignment" ayanamsha,
where the galactic equator cuts the ecliptic at exactly 0° Sagittarius. This
ayanamsha differs from the Skydram ayanamsha by only 19 arc seconds.

32)
Galactic Equator (Node) at 0° Sagittarius

The last
two ayanamshas are based on a slightly outdated position of the galactic pole
that was determined in 1958. According to more recent observations and
calculations from the year 2010, the galactic node with the

ecliptic
shifts by 3'11", and the "Galactic Alignment" is preponed to
1994. The galactic node is fixed exactly at sidereal 0° Sagittarius.

(= Galactic equator cuts ecliptic in the middle
of Mula and the beginning of Ardra)

With this
ayanamsha, the galactic equator cuts the ecliptic exactly in the middle of the
nakshatra Mula. This means that the Milky Way passes through the middle of this
lunar mansion. Here again, it is interesting that the Sanskrit word mūlam means "root,
origin", and it seems that the circle of the lunar mansions originally
began with this nakshatra. On the opposite side, the galactic equator cuts the
ecliptic exactly at the beginning of the nakshatra Ārdrā ("the moist, green, succulent one",
feminine).

This
ayanamsha was introduced by the American astrologer Ernst Wilhelm in 2004. He
used a calculation of the galactic node by D. Koch from the year 2001, which had
a small error of 2 arc seconds. The current implementation of this ayanamsha is
based on a new position of the Galactic pole found by Chinese

With this
ayanamsha, the star Mula (λ Scorpionis) is assumed at 0°
Sagittarius.

The Indian
astrologer Chandra Hari is of the opinion that the lunar mansion Mula
corresponds to the Muladhara Chakra. He refers to the doctrine of the Kalapurusha which assigns the 12 zodiac
signs to parts of the human body. The initial point of Aries is considered to
correspond to the crown and Pisces to the feet of the cosmic human being. In
addition, Chandra Hari notes that Mula has the advantage to be located near the
galactic centre and to have “no proper motion”. This ayanamsha is very close to
the Fagan/Bradley ayanamsha. Chandra Hari believes it defines the original
Babylonian zodiac.

(In
reality, however, the star Mula (λ Scorpionis) has a small proper motion, too.
As has been stated, the position of the galactic centre was not known to the
ancient peoples. However, they were aware of the fact that the Milky Way crossed
the ecliptic in this region of the sky.)

Sources:

- K.
Chandra Hari, "On the Origin of Siderial Zodiac and Astronomy", in: Indian Journal of History of Science,
33(4) 1998.

The
following ayanamshas were provided by Graham Dawson (”Solar Fire”), who had
taken them over from Robert Hand’s Program ”Nova”. Some were also contributed
by David Cochrane. Explanations by D. Koch:

2) De Luce
Ayanamsha

This
ayanamsha was proposed by the American astrologer Robert DeLuce (1877-1964). It
is fixed at the birth of Jesus, theoretically at 1 January 1 AD. However,
DeLuce de facto used an ayanamsha of 26°24'47 in the year 1900, which
corresponds to 4 June 1 BC as zero ayanamsha date.

DeLuce
believes that this ayanamsha was also used in ancient India. He draws this
conclusion from the fact that the important ancient Indian astrologer Varahamihira,
who assumed the solstices on the ingresses of the Sun into

sidereal
Cancer and Capricorn, allegedly lived in the 1st century BC. This dating of
Varahamihira has recently become popular under the influence of Hindu nationalist
ideology (Hindutva). However, historically, it cannot be

This
ayanamsha was used by the great Indian astrologer Bangalore Venkata Raman (1912-1998).
It is based on a statement by the medieval astronomer Bhaskara II (1184-1185),
who assumed an ayanamsha of 11° in the year 1183 (according to Information
given by Chandra Hari, unfortunately without giving his source). Raman himself
mentioned the year 389 CE as year of zero ayanamsha in his book Hindu Predictive Astrology, pp. 378-379.

Although
this ayanamsha is very close to the galactic ayanamsha of Gil Brand, Raman
apparently did not think of the possibility to define the zodiac using the
galactic centre.

This ayanamsha was allegedly recommended by Swami Shri
Yukteshwar Giri (1855-1936). We have taken over its definition from Graham
Dawson. However, the definition given by Yukteshwar himself in the introduction
of his work The Holy Science is a
confusing. According to his “astronomical reference books”, the ayanamsha on
the spring equinox 1894 was 20°54’36”. At the same time he believed that this
was the distance of the spring equinox from the star Revati, which he put at
the initial point of Aries. However, this is wrong, because on that date,
Revati was actually 18°24’ away from the vernal point. The error is explained
from the fact that Yukteshwar used the zero ayanamsha year 499 CE and an
inaccurate Suryasiddhantic precession rate of 360°/24’000 years = 54
arcsec/year. Moreover, Yukteshwar is wrong in assigning the above-mentioned
ayanamsha value to the year 1894; in reality it applies to 1893.

Since Yukteshwar’s precession rate is wrong by 4” per
year, astro.com cannot reproduce his horoscopes accurately for epochs far from
1900. In 2000, the difference amounts to 6’40”.

Although this ayanamsha differs only a few arc seconds
from the galactic ayanamsha of Gil Brand, Yukteshwar obviously did not intend
to define the zodiac using the galactic centre. He actually intended a
Revati-oriented

ayanamsha, but committed the above-mentioned errors in
his calculation.

This
ayanamsha was used by the Indian astrologer J.N. Bhasin (1908-1983).

6)
Djwhal Khul Ayanamsha

This ayanamsha is based on the assumption that the Age
of Aquarius will begin in the year 2117. This assumption is maintained by a
theosophical society called Ageless
Wisdom, and bases itself on a channelled message given in 1940 by a certain
spiritual master called Djwhal Khul.

Graham Dawson commented it as follows (E-mail to Alois
Treindl of 12 July 1999): ”The "Djwhal Khul" ayanamsha originates
from information in an article in the Journal of Esoteric Psychology, Volume
12, No 2, pp91-95, Fall 1998-1999 publ. Seven Ray Institute). It is based on an
inference that the Age of Aquarius starts in the year 2117. I decided to use
the 1st of July simply to minimise the possible error given that an exact date
is not given.”

Sources:

- Philipp Lindsay, “The Beginning of the Age of
Aquarius: 2,117 A.D.”, http://esotericastrologer.org/newsletters/the-age-of-aquarius-ray-and-zodiac-cycles/

In all
software known to us, sidereal planetary positions are computed from the
following equation:

sidereal_position
= tropical_position – ayanamsha,

The ayanamhsa
is computed from the ayanamsha(t0) at a starting date (e.g. 1 Jan 1900)
and the speed of the vernal point, the so-called precession rate.

This
algorithm is unfortunately too simple. At best, it can be considered an
approximation. The precession of the equinox is not only a matter of ecliptical
longitude, but is a more complex phenomenon. It has two components:

a) The soli-lunarprecession: The combined gravitational pull of the Sun and the Moon on
the equatorial bulge of the earth causes the earth to spin like a top. As a
result of this movement, the vernal point moves around the ecliptic with a
speed of about 50” per year. This cycle has a period of about 26000 years.

b) The planetary
precession: The earth orbit itself is not fixed. The gravitational
influence from the planets causes it to wobble. As a result, the obliquity of
the ecliptic currently decreases by 47” per century, and this change has an
influence on the position of the vernal point, too.

(Note, the rotation
pole of the earth is very stable, it the equator keeps an almost constant angle
relative to the ecliptic of a fixed date, with a change of only a couple of
0.06” cty-2.)

Because the
ecliptic is not fixed, it is not completely correct to subtract an ayanamsha
from the tropical position in order to get a sidereal position. Let us take,
e.g., the Fagan/Bradley ayanamsha, which is defined by:

ayanamsha =
24°02’31.36” + d(t)

24°02’31.36” is the value of the ayanamsha on 1 Jan
1950. It is obviously measured on the ecliptic of 1950.

d(t) is the distance of the vernal point
at epoch t from the position of the vernal point on 1 Jan 1950. However,
the whole ayanamsha is subtracted from a planetary position which is
referred to the ecliptic of the epoch t. This does not make sense. The ecliptic
of the epoch t0and the epoch t are not
exactly the same plane.

As a result,
objects that do not move sidereally, still do seem to move. If we compute its
precise tropical position for several dates and then subtract the Fagan/Bradley
ayanamsha for the same dates in order to get its sidereal position,
these positions will all be different. This can be considerable over long
periods of time:

Long-term ephemeris of some fictitious star near
the ecliptic that has no proper motion:

DateLongitudeLatitude

01.01.-12000335°16'55.22110°55'48.9448

01.01.-11000335°16'54.91390°47'55.3635

01.01.-10000335°16'46.59760°40'31.4551

01.01.-9000335°16'32.68220°33'40.6511

01.01.-8000335°16'16.22490°27'23.8494

01.01.-7000335°16' 0.18410°21'41.0200

01.01.-6000335°15'46.83900°16'32.9298

01.01.-5000335°15'37.45540°12' 1.7396

01.01.-4000335°15'32.22520° 8'10.3657

01.01.-3000335°15'30.45350° 5' 1.3407

01.01.-2000335°15'30.9235 0° 2'35.9871

01.01.-1000335°15'32.32680° 0'54.2786

01.01.0335°15'33.6425-0° 0' 4.7450

01.01.1000335°15'34.3645-0° 0'22.4060

01.01.2000335°15'34.5249-0° 0' 0.0196

01.01.3000335°15'34.52160° 1' 1.1573

Long-term
ephemeris of some fictitious star with high ecliptic latitude and no proper
motion:

DateLongitudeLatitude

01.01.-1200025°48'34.981258°55'17.4484

01.01.-1100025°33'30.570958°53'56.6536

01.01.-1000025°18'18.105858°53'20.5302

01.01.-900025° 3' 9.251758°53'26.8693

01.01.-800024°48'12.632058°54'12.3747

01.01.-700024°33'33.626758°55'34.7330

01.01.-600024°19'16.332558°57'33.3978

01.01.-500024° 5'25.484459° 0' 8.8842

01.01.-400023°52' 6.945759° 3'21.4346

01.01.-300023°39'26.868959° 7'10.0515

01.01.-200023°27'30.509859°11'32.3495

01.01.-100023°16'21.608159°16'25.0618

01.01.023°
6' 2.632459°21'44.7241

01.01.100022°56'35.564959°27'28.1195

01.01.200022°48'
2.625459°33'32.3371

01.01.300022°40'26.478659°39'54.5816

Exactly the
same kind of thing happens to sidereal planetary positions, if one calculates
them in the traditional way. The “fixed zodiac” is not really fixed.

The wobbling of the ecliptic plane also influences ayanamshas that are
referred to the nodes of the galactic equator with the ecliptic.

2)fixed-star-bound
ecliptic (implemented in Swiss Ephemeris for some selected stars)

One could
use a stellar object as an anchor for the sidereal zodiac, and make sure that a
particular stellar object is always at a certain position on the ecliptic of
date. E.g. one might want to have Spica always at 0 Libra or the Galactic
Center always at 0 Sagittarius. There is nothing against this method from a
geometrical point of view. But it has to be noted, that this system is not
really fixed either, because it is still based on the moving ecliptic, and
moreover the fixed stars have a small proper motion, as well. Thus, if Spica is
assumed as a fixed point, then all other stars will move even faster, because
the proper motion of Spica will be added to their own proper motions. Note, the
Galactic Centre (Sgr A*) is not really fixed either, but has a small apparent
motion that reflects the motion of the Sun about it.

This
solution has been implemented for the following stars and fixed postions:

Polar longitude of
galactic centre in the middle of nakshatra Mula (E. Wilhelm)

With Swiss
Ephemeris versions before 2.05, the apparent
position of the star relative to the mean
ecliptic plane of date was used as the reference point of the zodiac. E.g. with
the True Chitra ayanamsha, the star Chitra/Spica had the apparent position 180°
exactly. However, the true position
was slightly different. Since version 2.05, the star is always exactly at 180°,
not only its apparent, but also its true position.

A special
case are the ayanamshas that are based on the galactic node (the intersection
of the galactic equator with the mean ecliptic of date). These ayanamshas
include:

Galactic equator (IAU 1958)

Galactic
equator true/modern

Galactic equator in middle of Mula

(Note, the
Mardyks ayanamsha, although derived from the galactic equator, does not work
like this. It is calculated using the method described above under 1).)

The node is
calculated from the true position of
the galactic pole, not the apparent one. As a result, if the position of the
galactic pole is calculated with the ayanamsha that has the galactic node at 0°
Sagittarius, then the true position of the pole is exactly at sidereal 150°,
but its apparent position is slightly different from that.

3)projection
onto the ecliptic of t0 (implemented in Swiss Ephemeris as an option)

Another
possibility would be to project the planets onto the reference ecliptic of the ayanamsha
– for Fagan/Bradley, e.g., this would be the ecliptic of 1950 – by a correct coordinate
transformation and then subtract 24.042°, the initial value of the ayanamsha.

If we
follow this method, the effect described above under 1) (traditional ayanamsha
method) will not occur, and an object that has no proper motion will keep its
position forever.

This method
is geometrically more correct than the traditional one, but still has a
problem. For, if we want to refer everything to the ecliptic of some initial
date t0, we will have to choose that date very carefully. Its ecliptic ought to
be of special importance. The ecliptic of 1950 or the one of 1900 are obviously
meaningless and not suitable as a reference plane. So, how about some
historical date on which the tropical and the sidereal zero point coincided?
Although this may be considered as a kind of cosmic anniversary (the Sassanians
did so), the ecliptic plane of that time does not have an “eternal” value. It
is different from the ecliptic plane of the previous anniversary around the
year 26000 BC, and it is also different from the ecliptic plane of the next
cosmic anniversary around the year 26000 AD.

This
algorithm is supported by the Swiss Ephemeris, too. However, it must not be
used with the Fagan/Bradley definition or with other definitions that were
calibrated with the traditional method of ayanamsha subtraction. It can
be used for computations of the following kind:

a)Astronomers may want to calculate positions
referred to a standard equinox like J2000, B1950, or B1900, or to any other
equinox.

b)Astrologers may be interested in the
calculation of precession-corrected transits. See explanations in the
next chapter.

c)The algorithm can be applied to any
user-defined sidereal mode, if the ecliptic of its reference date is considered
to be astrologically significant.

d)The algorithm makes the problems of the
traditional method visible. It shows the dimensions of the inherent inaccuracy
of the traditional method. (Calculate some star position using the traditional
method and compare it to the same star’s position if calculated using this method.)

For the
planets and for centuries close to t0, the difference from the traditional
procedure will be only a few arc seconds in longitude. Note that the Sun will
have an ecliptical latitude of several arc minutes after a few centuries.

For the
lunar nodes, the procedure is as follows:

Because the
lunar nodes have to do with eclipses, they are actually points on the ecliptic
of date, i.e. on the tropical zodiac. Therefore, we first compute the nodes as
points on the ecliptic of date and then project them onto the sidereal zodiac.
This procedure is very close to the traditional method of computing sidereal
positions (a matter of arc seconds). However, the nodes will have a latitude of
a couple of arc minutes.

For the
axes and houses, we compute the points where the horizon or the house lines
intersect with the sidereal plane of the zodiac, not with the ecliptic
of date. Here, there are greater deviations from the traditional procedure. If t
is 2000 years from t0, the difference between the ascendant positions
might well be 1/2 degree.

To avoid
the problem of choice of a reference ecliptic, one could use a kind of ”average
ecliptic”. The mechanism of the planetary precession mentioned above works in a
similar way as the mechanism of the luni-solar precession. The motion of the
earth orbit can be compared to a spinning top, with the earth mass equally
distributed around the whole orbit. The other planets, especially Venus and
Jupiter, cause it to move around an average position. But unfortunately, this “long-term
mean Earth-Sun plane” is not really stable either, and therefore not suitable
as a fixed reference frame.

The period
of this cycle is about 75000 years. The angle between the long-term mean plane
and the ecliptic of date is currently about 1°33’, but it varies considerably.
(This cycle must not be confused with the period between two maxima of the
ecliptic obliquity, which is about 40000 years and often mentioned in the
context of planetary precession. This is the time it takes the vernal point to
return to the node of the ecliptic (its rotation point), and therefore the
oscillation period of the ecliptic obliquity.)

5)The
solar system rotation plane (implemented in Swiss Ephemeris as an option)

The solar
system as a whole has a rotation axis, too, and its equator is quite close to
the ecliptic, with an inclination of 1°34’44” against the ecliptic of the year
2000. This plane is extremely stable and probably the only convincing candidate
for a fixed zodiac plane.

This method
avoids the problem of method 3). No particular ecliptic has to be chosen as a
reference plane. The only remaining problem is the choice of the zero point.

It does not
make much sense to use this algorithm for predefined sidereal modes. One can
use this algorithm for user-defined ayanamshas.

Method no.
3, the transformation to the ecliptic of t0, opens two more possibilities:

You can
compute positions referred to any equinox, 2000, 1950, 1900, or whatever you
want. This is sometimes useful when Swiss Ephemeris data ought to be compared
with astronomical data, which are often referred to a standard equinox.

There are
predefined sidereal modes for these standard equinoxes:

18) J2000

19) J1900

20) B1950

Moreover,
it is possible to compute precession-corrected transits or synastries
with very high precision. An astrological transit is defined as the passage of
a planet over the position in your birth chart. Usually, astrologers assume
that tropical positions on the ecliptic of the transit time has to be compared
with the positions on the tropical ecliptic of the birth date. But it has been
argued by some people that a transit would have to be referred to the ecliptic
of the birth date. With the new Swiss Ephemeris algorithm (method no. 3) it is
possible to compute the positions of the transit planets referred to the
ecliptic of the birth date, i.e. the so-called precession-corrected
transits. This is more precise than just correcting for the precession in
longitude (see details in the programmer's documentation swephprg.doc, ch.
8.1).

The Swiss ephemeris provides the calculation of apparent or true
planetary positions. Traditional astrology works with apparent positions.
”Apparent” means that the position where we see the planet is used, not
the one where it actually is. Because the light's speed is finite, a planet is
never seen exactly where it is. (see above, 2.1.3 ”The details of coordinate
transformation”, light-time and aberration) Astronomers therefore make a
difference between apparent and true positions. However, this
effect is below 1 arc minute.

Most astrological ephemerides provide apparent positions.
However, this may be wrong. The use of apparent positions presupposes that
astrological effects can be derived from one of the four fundamental forces of
physics, which is impossible. Also, many astrologers think that astrological
”effects” are a synchronistic phenomenon (the ones familiar with physics may
refer to the Bell theorem). For such reasons, it might be more convincing to
work with true positions.

Moreover, the Swiss Ephemeris supports so-called astrometric
positions, which are used by astronomers when they measure positions of
celestial bodies with respect to fixed stars. These calculations are of no use
for astrology, though.

More precisely speaking, common ephemerides tell us the position where
we would see a planet if we stood in the center of the earth and could see the
sky. But it has often been argued that a planet’s position ought to be referred
to the geographic position of the observer (or the birth place). This can make
a difference of several arc seconds with the planets and even more than a
degree with the moon! Such a position referred to the birth place is called
the topocentric planetary position. The observation of transits over the
moon might help to find out whether or not this method works better.

For very precise topocentric calculations, the Swiss Ephemeris not only
requires the geographic position, but also its altitude above sea. An altitude
of 3000 m (e.g. Mexico City) may make a difference of more than 1 arc second
with the moon. With other bodies, this effect is of the amount of a 0.01”. The
altitudes are referred to the approximate earth ellipsoid. Local irregularities
of the geoid have been neglected.

Our topocentric lunar positions differ from the NASA positions (s. the Horizons
Online Ephemeris System http://ssd.jpl.nasa.gov) by 0.2 - 0.3 arc sec. This
corresponds to a geographic displacement by a few 100 m and is about the best
accuracy possible. In the documentation of the HorizonsSystem,
it is written that: "The Earth is assumed to be a rigid body. ... These
Earth-model approximations result in topocentric station location errors, with
respect to the reference ellipsoid, of less than 500 meters."

The Swiss ephemeris also allows the computation of apparent or true topocentric
positions.

With the lunar nodes and apogees, Swiss Ephemeris does not make a
difference between topocentric and geocentric positions. There are manyfold
ways to define these points topocentrically. The simplest one is to understand
them as axes rather than points somewhere in space. In this case, the
geocentric and the topocentric positions are identical, because an axis is an
infinite line that always points to the same direction, not depending on the
observer's position.

Moreover, the Swiss Ephemeris supports the calculation of heliocentric
and barycentric planetary positions. Heliocentric positions are positions
as seen from the center of the sun rather than from the center of the earth.
Barycentric ones are positions as seen from the center of the solar system,
which is always close to but not identical to the center of the sun.

From Swiss Ephemeris version 1.76 on, heliacal
events have been included. The heliacal rising and setting of planets and stars
was very important for ancient Babylonian and Greek astronomy and
astrology.Also, first and last
visibility of the Moon can be calculated, which are important for many
calendars, e.g. the Islamic, Babylonian and ancient Jewish calendars.

The heliacal events that can be determined are:

·Inferior planets

·Heliacal rising (morning first)

·Heliacal setting (evening last)

·Evening first

·Morning
last

·Superior planets and stars

·Heliacal rising

·Heliacal
setting

·Moon

·Evening first

·Morning last

The
acronychal risings and settings (also called cosmical settings) of superior
planets are a different matter. They will be added in a future version of the
Swiss Ephemeris.

The principles behind the calculation are based
on the visibility criterion of Schaefer [1993, 2000], which includes
dependencies on aspects of:

·Position
celestial objects
like the position and magnitude of the Sun, Moon and the studied celestial
object,

·Contrast
between studied object and sky background
The observer’s eye can on detect a certain amount of contract and this contract
threshold is the main body of the calculations

In the following sections above aspects will be
discussed briefly and an idea will be given what functions are available to
calculate the heliacal events. Lastly the future developments will be
discussed.

The theory behind this visibility criterion is
explained by Schaefer [1993, 2000] and the implemented by Reijs [2003] and Koch
[2009]. The general ideas behind this theory are explained in the following
subsections.

To determine the visibility of a celestial
object it is important to know where the studied celestial object is and what
other light sources are in the sky. Thus beside determining the position of the
studied object and its magnitude, it also involves calculating the position of
the Sun (the main source of light) and the Moon. This is common functions
performed by Swiss Ephemeris.

The location of the observer determines the
topocentric coordinates (incl. influence of refraction) of the celestial object
and his/her height (and altitude of studied object) will have influence on the
amount of airmass that is in the path of celestial object’s light.

The observer’s eyes will determine the
resolution and the contrast differences he/she can perceive (depending on age
and acuity), furthermore the observer might used optical instruments (like
binocular or telescope).

The meteorological circumstances are very
important for determining the visibility of the celestial object. These
circumstances influence the transparency of the airmass (due to
Rayleigh&aerosol scattering and ozone&water absorption) between the
celestial object and the observer’s eye. This result in the astronomical
extinction coefficient (AEC: ktot). As this is a complex
environment, it is sometimes ‘easier’ to use a certain AEC, instead of
calculating it from the meteorological circumstances.

All the above aspects influence the perceived
brightnesses of the studied celestial object and its background sky. The
contrast threshold between the studied object and the background will determine
if the observer can detect the studied object.

In general this procedure works in such a way
that it finds (through iterations) the day that conjunction/opposition between
Sun and studied object happens and then it determines, close to Sun’s rise or
set (depending on the heliacal event), when the studied object is visible (by
using the swe_vis_limit_magn function).

This function is limited to geographic
latitudes between 60S and 60N. Beyond that the heliacal phenomena of the
planets become erratic.We found cases of
strange planetary behavior even at 55N.

An example:

At 0E, 55N, with an extinction coefficient 0.2,
Mars had a heliacal rising on 25 Nov. 1957. But during the following weeks and
months Mars did not constantly increase its height above the horizon before
sunrise. In contrary, it disappeared again, i.e. it did a “morning last” on 18
Feb. 1958. Three months later, on 14 May 1958, it did a second morning first
(heliacal rising). The heliacal setting or evening last took place on 14 June
1959.

Currently, this special case is not handled by
swe_heliacal_ut(). The function cannot detect “morning lasts” of Mars and
following “second heliacal risings”. The function only provides the heliacal
rising of25 Nov. 1957 and the next
ordinary heliacal rising in its ordinary synodic cycle which took place on 11
June 1960.

However, we may find a solution for such
problems in future releases of the Swiss Ephemeris and even extend the
geographic range of swe_heliacal_ut() to beyond +/- 60.

For the Moon, swe_heliacal_ut() calculates the
evening first and the morning last. For each event, the function returns the
optimum visibility and a time of visibility start and visibility end. Note,
that on the day of its morning last or evening first, the moon is often visible
for almost the whole day. It even happens that the crescent first becomes
visible in the morning after its rising, then disappears, and again reappears
during culmination, because the observation conditions are better as the moon
stands high above the horizon. The function swe_heliacal_ut() does not handle
this in detail. Even if the moon is visible after sunrise, the function assumes
that the end time of observation is at sunrise. The evening fist is handled in
the same way.

With Venus, we have a similar situation. Venus
is often accessible to naked eye observation during day, and sometimes even
during inferior conjunction, but usually only at a high altitude above the
horizon. This means that it may be visible in the morning at its heliacal
rising, then disappear and reappear during culmination.(Whoever does not believe me, please read the
article by H.B. Curtis listed under “References”.)The function swe_heliacal_ut() does not
handle this case. If binoculars or a telescope is used, Venus may be even
observable during most of the day on which it first appears in the east.

The section of the Swiss Ephemeris software is
still under development. The acronychal events of superior planets and stars
will be added. And comparing other visibility criterions will be included; like
Schoch’s criterion [1928], Hoffman’s overview [2005] and
Caldwall&Laney criterion [2005].

- Schoch, K., Astronomical and calendrical
tables in Langdon. S., Fotheringham, K.J., The Venus tables of Amninzaduga:
A solution of Babylonian chronology by means of Venus observations of the first
dynasty, Oxford, 1928.

Swiss Ephemeris
versions until 1.80 used the IAU 1976 formula for Sidereal time. Since version
2.00 it uses sidereal time based on the IAU2006/2000 precession/nutation model.

As this solution
is not good for the whole time range of JPL Ephemeris DE431, we only use it
between 1850 and 2050. Outside this period, we use a solution based on the long
term precession model Vondrak 2011, nutation IAU2000 and the mean motion of the
Earth according to Simon & alii 1994. To make the function contiuous we add
some constant values to our long-term function before 1850 and after 2050.

Vondrak/Capitaine/Wallace,
"New precession expressions, valid for long time intervals", in
A&A 534, A22(2011).

This system is named after the Italian monk
Placidus de Titis (1590-1668). The cusps are defined by divisions of
semidiurnal and seminocturnal arcs. The 11th
cusp is the point on the ecliptic that has completed 2/3 of its semidiurnal
arc, the 12th cusp the point that has completed 1/3 of it.
The 2nd cusp has completed 2/3 of its seminocturnal arc, and the 3rd cusp 1/3.

This system is called after
the German astrologer Walter Koch (1895-1970). Actually it was invented by
Fiedrich Zanzinger and Heinz Specht, but it was made known by Walter Koch. In
German-speaking countries, it is also called the "Geburtsorthäusersystem"
(GOHS), i.e. the "Birth place house system". Walter Koch thought that
this system was more related to the birth place than other systems, because all
house cusps are computed with the "polar height of the birth place",
which has the same value as the geographic latitude.

This argumentation shows
actually a poor understanding of celestial geometry. With the Koch system, the
house cusps are actually defined by horizon lines at different times. To
calculate the cusps 11 and 12, one can take the time it took the MC degree to
move from the horizon to the culmination, divide this time into three and see
what ecliptic degree was on the horizon at the thirds. There is no reason why
this procedure should be more related to the birth place than other house
methods.

Named after the Johannes
Müller (called "Regiomontanus", because he stemmed from Königsberg)
(1436-1476).

The equator is divided into 12
equal parts and great circles are drawn through these divisions and the north
and south points on the horizon. The intersection points of these circles with
the ecliptic are the house cusps.

The vertical great circle from
east to west is divided into 12 equal parts and great circles are drawn through
these divisions and the north and south points on the horizon. The intersection
points of these circles with the ecliptic are the house cusps.

The first house starts at the
beginning of the zodiac sign in which the ascendant is. Each house covers a
complete sign. This method was used in Hellenistic astrology and is still used
in Hindu astrology.

This is a traditional Indian house system. In a first step, Porphyry
houses are calculated. The cusps of each new house
will be the midpoint between the last and the current. So house 1 will be equal
to:

This house system was invented in 1994 by Walter Pullen, the author of
the astrology software Astrolog. Like the Porphyry house system, this house system
is based on the idea that the division of the houses must be relative tothe ecliptic
sections of the quadrants only, not relative to the equator or diurnal
arcs. For this reason, Pullen originally called it “Neo-Porphyry”. However, the
sizes of the houses of a quadrant are not equal. Pullen describes it as
follows:

“Like Porphyry except instead of simply
trisecting quadrants, fit the house widths to a sine wave such that the
2nd/5th/8th/11th houses are expanded or compressed more based on the relative size
of their quadrants.”

In practice, an ideal house size of 30° each is assumed, then half of
the deviation of the quadrant from 90° is added to the middle house of the
quadrant. The remaining half is bisected again into quarters, and a quarter is
added to each of the remaining houses. Pullen himself puts it as follows:

In January 2016, in a
discussion in the Swiss Ephemeris Yahoo Group, Alois Treindl criticised that
Pullen’s code only worked as long as the quadrants were greater than 30°,
whereas negative house sizes resulted for the middle house of quadrants smaller
than 30°. It was agreed upon that in such cases the size of the house had to be
set to 0.

On 24 Jan. 2016, during the
above-mentioned discussion in the Swiss Ephemeris Yahoo Group, Walter Pullen
proposed a better solution of a sinusoidal division of the quadrants, which
does not suffer from the same problem. He wrote:

“It's
possible to do other than simply add sine wave offsets to houses (the "Sinusoidal
Delta" house system above). Instead, let's proportion or ratio the entire
house sizes themselves to each other based on the sine wave constants, or
multiply instead of add. That results in using a "sinusoidal ratio"
instead of a "sinusoidal delta", so this alternate method could be
called "Sinusoidal Ratio houses". As before, let "x" be the
smallest house in the compressed quadrant. There's a ratio "r"
multiplied by it to get the slightly larger 10th and 12th houses. The value
"r" starts out at 1.0 for 90 degree quadrants, and gradually
increases as quadrant sizes differ. Houses in the large quadrant have
"r" multiplied to "x" 3 times (or 4 times for the largest
quadrant). That results in the (0r, 1r, 3r, 4r) distribution from the sine wave
above. This is summarized in the chart below:”

The equator is partitioned
into 12 equal parts starting from the ARMC. Then the ecliptic points are
computed that have these divisions as their right ascension. Note: The
ascendant is different from the 1st house cusp.

The equator is partitioned
into 12 equal parts starting from the right ascension of the ascendant. Then
the ecliptic points are computed that have these divisions as their right
ascension. Note: The MC is different from the 10th house cusp.

The prefix “poli-“ might stand
for “polar”. (Speculation by DK.)

Carter’s own words:

“...the houses are demarcated
by circles passing through the celestial poles and dividing the equator into
twelve equal arcs, the cusp of the 1st house passing through the ascendant.
This system, therefore, agrees with the natural rotation of the heavens and
also produces, as the Ptolemaic (equal) does not, distinctive cusps for each
house....”

The equator is divided into 12
equal parts starting from the ARMC. The resulting 12 points on the equator are
transformed into ecliptic coordinates. Note: The Ascendant is different from
the 1st cusp, and the MC is different from the 10th cusp.

This system was introduced in
1961 by Wendel Polich and A.P. Nelson Page. Its construction is rather
abstract: The tangens of the polar height of the 11th house is the
tangens of the geo. latitude divided by 3. (2/3 of it are taken for the 12th
house cusp.) The philosophical reasons for this algorithm are obscure. Nor is
this house system more “topocentric” (i.e. birth-place-related) than any other
house system. (c.f. the misunderstanding with the “birth place system”.) The
“topocentric” house cusps are close to Placidus house cusps except for high
geographical latitudes. It also works for latitudes beyond the polar circles,
wherefore some consider it to be an improvement of the Placidus system.
However, the striking philosophical idea behind Placidus, i.e. the division of
diurnal and nocturnal arcs of points of the zodiac, is completely abandoned.

A method of house division
which first appears with the Hellenistic astrologer Rhetorius (500 A.D.) but is
named after Alcabitius, an Arabic astrologer, who lived in the 10th century
A.D. This is the system used in a few remaining examples of ancient Greek
horoscopes.

The MC and ASC are the 10th-
and 1st- house cusps. The remaining cusps are determined by the trisection of
the semidiurnal and seminocturnal arcs of the ascendant, measured on the
celestial equator. The houses are formed by great circles that pass through
these trisection points and the celestial north and south poles.

This is the “house” system
used by the Gauquelins and their epigones and critics in statistical
investigations in Astrology. Basically, it is identical with the Placidus house
system, i.e. diurnal and nocturnal arcs of ecliptic points or planets are
subdivided. There are a couple of differences, though.

-Up to 36 “sectors” (or house cusps) are used instead
of 12 houses.

-The sectors are counted in clockwise direction.

-There are so-called plus (+) and minus (–) zones. The
plus zones are the sectors that turned out to be significant in statistical
investigations, e.g. many top sportsmen turned out to have their Mars in a plus
zone. The plus sectors are the sectors 36 – 3, 9 – 12, 19 – 21, 28 – 30.

-More sophisticated algorithms are used to calculate
the exact house position of a planet (see chapters 6.4 and 6.5 on house
positions and Gauquelin sector positions of planets).

This house system was first published in 1994/1995 by three different
authors independently of each other:

- by Bogdan
Krusinski (Poland)

- by Milan Pisa
(Czech Republic) under the name “Amphora house system”.

- by Georg Goelzer (Switzerland) under the name “Ich-Kreis-Häusersystem”
(“I-Circle house system”)..

Krusinski
defines the house system as “… based on the great circle passing through
ascendant and zenith. This circle is divided into 12 equal parts (1st cusp is
ascendant, 10th cusp is zenith), then the resulting points are projected onto
the ecliptic through meridian circles. The house cusps in space are
half-circles perpendicular to the equator and running from the north to the
south celestial pole through the resulting cusp points on the house circle. The
points where they cross the ecliptic mark the ecliptic house cusps.”
(Krusinski, e-mail to Dieter Koch)

It
may seem hard to believe that three persons could have discovered the same
house system at almost the same time. But apparently this is what happened.
Some more details are given here, in order to refute wrong accusations from the
Czech side against Krusinski of having “stolen” the house system.

Out of the documents that Milan Pisa sent to
Dieter Koch, the following are to be mentioned: Private correspondence from
1994 and 1995 on the house system between Pisa and German astrologers Böer and
Schubert-Weller, two type-written (apparently unpublished) treatises in German
on the house system dated from 1994. A printed booklet of 16 pages in Czech
from 1997 on the theory of the house system (“Algoritmy noveho systemu
astrologickych domu”). House tables computed by Michael Cifka for the
geographical latitude of Prague, copyrighted from 1996. The house system was
included in the Czech astrology software Astrolog v. 3.2 (APAS) in 1998. Pisa’s
first publication on the house system happened in spring 1997 in “Konstelace“
No. 22, the periodical of the Czech Astrological Society.

Bogdan Krusinski first published the house
system at an astrological congress in Poland, the “"XIV
Szkola Astrologii Humanistycznej" held by Dr Leszek Weres, which took
place between 15.08.1995 and 28.08.1995 inPogorzelica at cost of the Baltic Sea.” Since
then Krusinski has distributed printed house tables for the Polish geographical
latitudes 49-55° and floppy disks with house tables for latitudes 0-90°. In
1996, he offered his program code to Astrodienst for implementation of this
house system into Astrodienst’s then astronomical software Placalc. (At that
time, however, Astrodienst was not interested in it.) In May 1997, Krusinski
put the data on the web at http://www.ci.uw.edu.pl/~bogdan/astrol.htm (now at
http://www.astrologia.pl/main/domy.html) In February 2006 he sent
Swiss-Ephemeris-compatible program code in C for this house system to
Astrodienst. This code was included into Swiss Ephemeris Version 1.70 and
released on 8 March 2006.

Georg Goelzer describes the same house system
in his book “Der Ich-Kosmos”, which appeared in July 1995 at Dornach,
Switzerland (Goetheanum). The book has a second volume with house tables
according to the house method under discussion. The house tables were created
by Ulrich Leyde. Goelzer also uses a house calculation programme which has the
time stamp “9 April 1993” and produces the same house cusps.

By March 2006, when the house system was
included in the Swiss Ephemeris under the nameof “Krusinski houses”, neither Krusinski nor Astrodienst knew about the
works of Pisa and Goelzer. Goelzer heard of his co-discoverers only in 2012 and
then contacted Astrodienst.

Conclusion: It seems that the house system was
first “discovered” and published byGoelzer, but that Pisa and Krusinski also “discovered” it independently.
We do not consider this a great miracle because the number of possible house
constructions is quite limited.

This house system was introduced by the Dutch astrologer L. Knegt and is
used by the Dutch Werkgemeenschap van Astrologen (WvA, also known as “Ram
school”).

The parallel of
declination that goes through the ascendant is divided in six equal parts both
above and below the horizon. Position circles through the north and the south
point on the horizon are drawn through he division points. The house cusps are
where the position circles intersect the ecliptic.

Note, the house
cusps 11, 12, 2, and 3 are not exactly opposite the cusps 5, 6, 8, and 9.

6.1.15.
Sunshine house system

The diurnal and nocturnal arcs of the Sun are trisected, and great
circles are drawn through these trisection points and the north and the south
point on the horizon. The intersection points of these great circles with the
ecliptic are the house cusps. Note that the cusps 11, 12, 2, and 3 are not in
exact opposition to the cusps 5, 6, 8, and 9.

For the polar region and during times where the Sun does not rise or
set, the diurnal and nocturnal arc are assumed to be either 180° or 0°. If the
diurnal arc is 0°, the house cusps 8 – 12 coincide with the meridian. If the
nocturnal arc is 0°, the cusps 2 – 6 coincide with the meridian. As with the
closely related Regiomontanus system, an MC below the horizon and IC above the
horizon are exchanged.

The Vertex is the point
of the ecliptic that is located precisely in western direction. The Antivertex
is the opposition point and indicates the precise east in the horoscope. It is
identical to the first house cusp in the horizon house system.

There is a lot of confusion
about this, because there is also another point which is called the "East
Point" but is usually not located in the east. In celestial
geometry, the expression "East Point" means the point on the horizon
which is in precise eastern direction. The equator goes through this point as
well, at a right ascension which is equal to ARMC + 90 degrees. On the other
hand, what some astrologers call the "East Point" is the point on the
ecliptic whose right ascension is equal to ARMC + 90 (i.e. the right ascension
of the horizontal East Point). This point can deviate from eastern direction by
23.45 degrees, the amount of the ecliptic obliquity. For this reason, the
term"East Point" is not very
well-chosen for this ecliptic point, and some astrologers (M. Munkasey) prefer
to call it the Equatorial Ascendant. This, because it is identical to
the Ascendant at a geographical latitude 0.

The Equatorial Ascendant is
identical to the first house cusp of the axial rotation system.

Note: If a projection of the
horizontal East Point on the ecliptic is wanted, it might seem more reasonable
to use a projection rectangular to the ecliptic, not rectangular to the equator
as is done by the users of the "East Point". The planets, as well,
are not projected on the ecliptic in a right angle to the ecliptic.

The Swiss Ephemeris supports
three more points connected with the house and angle calculation. They are part
of Michael Munkasey's system of the 8 Personal Sensitive Points (PSP).
The PSP include the Ascendant, the MC, the Vertex, the EquatorialAscendant, the AriesPoint, the LunarNode,
and the "Co-Ascendant" and the "PolarAscendant".

The term
"Co-Ascendant" seems to have been invented twice by two different
people, and it can mean two different things. The one "Co-Ascendant"
was invented by Walter Koch (?). To calculate it, one has to take the ARIC as
an ARMC and compute the corresponding Ascendant for the birth place. The
"Co-Ascendant" is then the opposition to this point.

The second "Co-Ascendant"
stems from Michael Munkasey. It is the Ascendant computed for the natal ARMC
and a latitude which has the value 90° - birth_latitude.

The "Polar
Ascendant" finally was introduced by Michael Munkasey. It is the
opposition point of Walter Koch's version of the "Co-Ascendant".
However, the "Polar Ascendant" is not the same as an Ascendant
computed for the birth time and one of the geographic poles of the earth. At
the geographic poles, the Ascendant is always 0 Aries or 0 Libra. This is not
the case for Munkasey's "Polar Ascendant".

1)We make sure that
the ascendant is always in the eastern hemisphere.

2)Placidus and Koch house
cusps sometimes can, sometimes cannot be computed beyond the polar circles.
Even the MC doesn't exist always, if one defines it in the Placidus manner. Our
function therefore automatically switches to Porphyry houses (each quadrant is
divided into three equal parts) and returns a warning.

3)Beyond the polar
circles, the MC is sometimes below the horizon. The geometrical definition of
the Campanus and Regiomontanus systems requires in such cases
that the MC and the IC are swapped. The whole house system is then oriented in
clockwise direction.

There are similar problems with the Vertex and the horizon
house system for birth places in the tropics. The Vertex is defined
as the point on the ecliptic that is located in precise western direction. The
ecliptic east point is the opposition point and is called the Antivertex.
Our program code makes sure that the Vertex (and the cusps 11, 12, 1, 2, 3 of
the horizon house system) is always located in the western hemisphere. Note
that for birthplaces on the equator the Vertex is always 0 Aries or 0 Libra.

Of course, there are no problems in the calculation of the Equatorial
Ascendant for any place on earth.

Placalc is the predecessor of Swiss Ephemeris; it is a calculation
module created by Astrodienst in 1988 and distributed as C source code. Beyond
the polar circles, Placalc‘s house calculation did switch to Porphyry houses
for all unequal house systems. Swiss Ephemeris still does so with the Placidus
and Koch method, which are not defined in such cases. However, the computation
of the MC and Ascendant was replaced by a different model in some cases: Swiss
Ephemeris gives priority to the Ascendant, choosing it always as the
eastern rising point of the ecliptic and accepting an MC below the horizon,
whereas Placalc gave priority to the MC. The MC was always chosen as the
intersection of the meridian with the ecliptic above the horizon. To
keep the quadrants in the correct order, i.e. have an Ascendant in the left
side of the chart, the Ascendant was switched by 180 degrees if necessary.

In the discussions between Alois Treindl and Dieter Koch during the
development of the Swiss Ephemeris it was recognized that this model is more
unnatural than the new model implemented in Swiss Ephemeris.

Placalc also made no difference between Placidus/Koch on one hand and
Regiomontanus/Campanus on the other as Swiss Ephemeris does. In Swiss
Ephemeris, the geometrical definition of Regiomontanus/Campanus is strictly
followed, even for the price of getting the houses in ”wrong” order. (see
above, chapter 4.1.)

b) ASTROLOG program as written by Walter Pullen

While the freeware program Astrolog contains the planetary routines of
Placalc, it uses its own house calculation module by Walter Pullen. Various
releases of Astrolog contain different approaches to this problem.

c) ASTROLOG on our website

ASTROLOG is also used on Astrodienst’s website for displaying free
charts. This version of Astrolog used on our website however is different from
the Astrolog program as distributed on the Internet. Our webserver version of
Astrolog contains calls to Swiss Ephemeris for planetary positions. For
Ascendant, MC and houses it still uses Walter Pullen's code. This will change
in due time because we intend to replace ASTROLOG on the website with our own
charting software.

d) other astrology programs

Because most astrology programs still use the Placalc module, they
follow the Placalc method for houses inside the polar circles. They give
priority to keep the MC above the horizon and switch the Ascendant by 180
degrees if necessary to keep the quadrants in order.

The Swiss Ephemeris DLL also provides a function to compute the house
position of a given body, i.e. in which house it is. This function can be used
either to determine the house number of a planet or to compute its position in
a house horoscope. (A house horoscope is a chart in which all
houses are stretched or shortened to a size of 30 degrees. For unequal house
systems, the zodiac is distorted so that one sign of the zodiac does not
measure 30 house degrees)

Note that the actual house position of a planet is not always the one
that it seems to be in an ordinary chart drawing. Because the planets are not
always exactly located on the ecliptic but have a latitude, they can seemingly
be located in the first house, but are actually visible above the horizon. In
such a case, our program function will place the body in the 12th (or 11 th or
10 th) house, whatever celestial geometry requires. However, it is possible to
get a house position in the ”traditional” way, if one sets the ecliptic
latitude to zero.

Although it is not possible to compute Placidus house cusps
beyond the polar circle, this function will also provide Placidus house
positions for polar regions. The situation is as follows:

The Placidus method works with the semidiurnal and seminocturnal arcs of
the planets. Because in higher geographic latitudes some celestial bodies (the
ones within the circumpolar circle) never rise or set, such arcs do not exist.
To avoid this problem it has been proposed in such cases to start the diurnal
motion of a circumpolar body at its ”midnight” culmination and its nocturnal
motion at its midday culmination. This procedure seems to have been proposed by
Otto Ludwig in 1930. It allows to define a planet's house position even if it
is within the circumpolar region, and even if you are born in the northernmost
settlement of Greenland. However, this does not mean that it be possible to
compute ecliptical house cusps for such locations. If one tried that, it would
turn out that e.g. an 11 th house cusp did not exist, but there were two 12th
house cusps.

Note however, that circumpolar bodies may jump from the 7th house
directly into the 12th one or from the 1st one directly into the 6th one.

The Koch method, on the other hand, cannot be helped even with
this method. For some bodies it may work even beyond the polar circle, but for
some it may fail even for latitudes beyond 60 degrees. With fixed stars, it may
even fail in central Europe or USA. (Dieter Koch regrets the connection of his
name with such a badly defined house system)

Note that Koch planets do strange jumps when the cross the meridian.
This is not a computation error but an effect of the awkward definition of this
house system. A planet can be east of the meridian but be located in the

9th house, or west of the meridian and in the 10th house. It is possible
to avoid this problem or to make Koch house positions agree better with the
Huber ”hand calculation” method, if one sets the ecliptic latitude of the
planets to zero. But this is not more correct from a geometrical point of view.

The calculation of the
Gauquelin sector position of a planet is based on the same idea as the Placidus
house system, i.e. diurnal and nocturnal arcs of ecliptic points or planets are
subdivided.

Three different algorithms
have been used by Gauquelin and others to determine the sector position of a
planet.

1.We can take the ecliptic point of the planet
(ecliptical latitude ignored) and calculate the fraction of its diurnal or
nocturnal arc it has completed

2.We can take the true planetary position (taking into
account ecliptical latitude) for the same calculation.

3.We can use the exact times for rise and set of the
planet to determine the ratio between the time the planet has already spent
above (or below) the horizon and its diurnal (or nocturnal) arc. Times of rise
and set are defined by the appearance or disappearance of the center of the planet’s
disks.

All three methods are
supported by the Swiss Ephemeris.

Methods 1 and 2 also work for
polar regions. The Placidus algorithm is used, and the Otto Ludwig method
applied with circumpolar bodies. I.e. if a planet does not have a rise and set,
the “midnight” and “midday” culminations are used to define its semidiurnal and
seminocturnal arcs.

With method 3, we don’t try to
do similar. Because planets do not culminate exactly in the north or south, a
planet can actually rise on the western part of the horizon in high geographic
latitudes. Therefore, it does not seem appropriate to use meridian transits as
culmination times. On the other hand, true culmination times are not always
available. E.g. close to the geographic poles, the sun culminates only twice a
year.

The computation of planets uses the so called Ephemeris Time (ET)
which is a completely regular time measure. Computations of sidereal time and
houses, on the other hand, depend on the rotation of the earth, which is not
regular at all. The time used for such purposes is called Universal Time (UT)
or Terrestrial Dynamic Time (TDT). It is an irregular time measure, and
is roughly identical to the time indicated by our clocks (if time zones are
neglected). The difference between ET and UT is called DT (”Delta T”), and
is defined as DT = ET – UT.

The earth's rotation decreases slowly, currently at the rate of about
0.5 – 1 second per year. Even worse, this decrease is irregular itself. It
cannot precisely predicted but only derived from star observations. The values
of DT achieved like this must be tabulated. However, this
table, which is published yearly by the Astronomical Almanac, starts only at
1620, about the time when the telescope was invented. For more remote
centuries, DT must be estimated from old eclipse records. The
uncertainty is in the range of hours for the year 3000 B.C. For future times, DT is estimated from
the current and the general changing rate, depending on whether a short-term or
a long-term extrapolation is intended.

NOTE:The DT algorithms have been
improved with the Swiss Ephemeris release 1.64 (Stephenson 1997), with release
1.72 (Morrison/Stephenson 2004) and 1.77 (Espenak & Meeus). These changes
result in significant changes of the ephemeris for remote historical dates, if
Universal Time is used.

The Swiss Ephemeris computes DT as follows.

1633 - today + a
couple of years:

The tabulated values of DT, in hundredths of a second,
were taken from the Astronomical Almanac page K8 and K9 and are yearly updated.

The DT function adjusts for a value of secular tidal
acceleration ndot that is consistent with the ephemeris used (LE430 has ndot =
-25.82 arcsec per century squared, LE405/406 has ndot = -25.826, ELP2000 and
DE200 ndot = -23.8946).

To change ndot, one can either redefine SE_TIDAL_DEFAULT in swephexp.h
or use the routine swe_set_tid_acc() before calling the Swiss Ephemeris.

For dates before 1600, the polynomials published by Espenak and Meeus
(2006) are used, with linear interpolation. They are based on an assumed value
of ndot = -26. The program adjusts for an ndot that is consistent with the
ephemeris used. These formulae include the long-term formula by
Morrison/Stephenson (2004, p. 332), which is used for epochs before -500.

future:

For the time after the last tabulated value, we use the formula of
Stephenson (1997; p. 507), with a modification that avoids a jump at the end of
the tabulated period. A linear term is added that makes a slow transition from
the table to the formula over a period of 100 years. (Need not be updated, when
table will be enlarged.)

Differences between
the old and new algorithms (before and after release 1.77):

Differences between
the old and new algorithms (before and after release 1.72):

yeardifference in seconds (new - old)

-3000-4127

-2000-2130

-1000-760

0-20

1000-30

160010

16190.5

16200

Differences between
the old and new algorithms (before and after release 1.64):

yeardifference in seconds (new - old)

-30002900

01200

160029

161960

1620-0.6

1700-0.4

1800-0.1

1900-0.02

1940-0.001

19500

20000

20202

210023

3000-400

In 1620, where the DT table of the Astronomical
Almanac starts, there was a jump of a whole minute in the old algorithms. The
new algorithms has no jumps anymore.

The smaller differences for the period 1620-1955, where we still use the
same data as before, is due to a correction in the tidal acceleration of the
moon, which now has the same value as is also used by JPL for their DT calculations.

Swiss Ephemeris is written in portable C and the same code is used for
creation of the 32-bit Windows DLL and the link library. All data files are
fully portable between different hardware architectures.

To build the DLLs, we use Microsoft Visual C++ version 5.0 (for 32-bit).

The DLL has been successfully used in the following programming
environments:

Visual C++ 5.0 (sample code
included in the distribution)

Visual Basic 5.0(sample code and
VB declaration file included)

Delphi 2 and Delphi 3 (32-bit, declaration file included)

As the number ofusers grows, our
knowledge base about the interface details between programming environments and
the DLL grows. All such information is added to the distributed Swiss Ephemeris
and registered users are informed via an email mailing list.

Earlier version up to version 1.61 supported 16-bit Windows programming.
Since then, 16-bit support has been dropped.

We give a short overview of the most important functions contained in
the Swiss Ephemeris DLL. The detailed description of the programming interface
is contained in the document swephprg.doc which is
distributed together with the file you are reading.

PLACALC, the predecessor of SWISSEPH, included several functions that we
do not need for SWISSEPH anymore. Nevertheless we include them again in our
DLL, because some users of our software may have taken them over and use them
in their applications. However, we gave them new names that were more
consistent with SWISSEPH.

PLACALC used angular measurements in centiseconds a lot; a centisecond
is 1/100 of an arc second. The C type CSEC or centisec is a 32-bit integer.
CSEC was used because calculation with integer variables was considerably
faster than floating point calculation on most CPUs in 1988, when PLACALC was
written.

In the Swiss Ephemeris we have dropped the use of centiseconds and use
double (64-bit floating point) for all angular measurements.

Placalc is a planetary calculation module which was made available by
Astrodienst since 1988 to other programmers under a source code license.
Placalc is less well designed, less complete and not as precise as the Swiss
Ephemeris module. However, many developers of astrological software have used
it over many years and like it. Astrodienst has used it internally since 1989
for a large set of application programs.

To simplify the introduction of Swiss Ephemeris in 1997 in Astrodienst's
internal operation, we wrote an interface module which translates all calls to
Placalc functions into Swiss Ephemeris functions, and translates the results
back into the format expected in the Placalc Application Interface (API).

This interface (swepcalc.c and swepcalc.h) is part of the source code distribution of Swiss Ephemeris; it is not
contained in the DLL. All new software should be written directly for the
SwissEph API, but porting old Placalc software is convenient and very simple with
the Placalc API.

The calculation of the apparent position of a planet involves a
relativistic effect, which is the curvature of space by the gravity field of
the Sun. This can also be described by a semi-classical algorithm, where the
photon travelling from the planet to the observer is deflected in the Newtonian
gravity field of the Sun, where the photon has a non-zero mass arising from its
energy. To get the correct relativistic result, a correction factor 2.0 must be
included in the calculation.

A problem arises when a planet disappears behind the solar disk, as seen
from the Earth. Over the whole 6000 year time span of the Swiss Ephemeris, it
happens often.

Planet

number of passes behind the Sun

Mercury

1723

Venus

456

Mars

412

Jupiter

793

Saturn

428

Uranus

1376

Neptune

543

Pluto

57

A typical occultation of a planet by the Solar disk, which has a
diameter of approx. _ degree, has a duration of about 12 hours. For the outer
planets it is mostly the speed of the Earth's movement which determines this
duration.

Strictly speaking, there is no apparent position of a planet when
it is eclipsed by the Sun. No photon from the planet reaches the observer's eye
on Earth. Should one drop gravitational deflection, but keep aberration and
light-time correction, or should one switch completely from apparent positions
to true positions for occulted planets? In both cases, one would come up with
an ephemeris which contains discontinuities, when at the moment of occultation
at the Solar limb suddenly an effect is switched off.

Discontinuities in the ephemeris need to be avoided for several reasons.
On the level of physics, there cannot be a discontinuity. The planet cannot
jump from one position to another. On the level of mathematics, a non-steady
function is a nightmare for computing any derived phenomena from this function,
e.g. the time and duration of an astrological transit over a natal body,
oran aspect of the planet.

Nobody seems to have handled this problem before in astronomical
literature. To solve this problem, we have used the following approach: We
replace the Sun, which is totally opaque for electromagnetic waves and not
transparent for the photons coming from a planet behind it, by a transparent
gravity field. This gravity field has the same strength and spatial
distribution as the gravity field of the Sun. For photons from occulted
planets, we compute their path and deflection in this gravity field, and from
this calculation we get reasonable apparent positions also for occulted
planets.

The calculation has been carried out with a semi-classical Newtonian
model, which can be expected to give the correct relativistic result when it is
multiplied with a correction factor 2. The mass of the Sun is mostly
concentrated near its center; the outer regions of the Solar sphere have a low
mass density. We used the a mass density distribution from the Solar standard
model, assuming it to have spherical symmetry (our Sun mass distribution m® is
from Michael Stix, The Sun, p. 47). The path of photons through this gravity
field was computed by numerical integration. The application of this model in
the actual ephemeris could then be greatly simplified by deriving an effective
Solar mass which a photon ”sees” when it passes close by or ”through” the Sun.
This effective mass depends only from the closest distance to the Solar center
which a photon reaches when it travels from the occulted planet to the
observer. The dependence of the effective mass from the occulted planet's
distance is so small that it can be neglected for our target precision of 0.001
arc seconds.

For a remote planet just at the edge of the Solar disk the gravity
deflection is about 1.8”, always pointing away from the center of the Sun. This
means that the planet is already slightly behind the Solar disk (with a
diameter of 1800”) when it appears to be at the limb, because the light bends
around the Sun. When the planet now passes on a central path behind the Solar
disk, the virtual gravity deflection we compute increases to 2.57 times the
deflection at the limb, and this maximum is reached at _ of the Solar radius.
Closer to the Solar center, the deflection drops and reaches zero for photons
passing centrally through the Sun's gravity field.

We have discussed our approach with Dr. Myles Standish from JPL and here
is his comment (private email to Alois Treindl, 12-Sep-1997):